The Ginzburg-Landau Equation Solved by the Finite Element Method
DEFF Research Database (Denmark)
Alstrøm, Tommy Sonne; Sørensen, Mads Peter; Pedersen, Niels Falsig
2006-01-01
Around 1950 V.L. Ginzburg and L.D. Landau proposed a phenomenological theory for phase transitions1. The theory is based on a phenomenological Schrödinger equation with a φ-4 potential and a kinetic term involving the momentum operator. One of the more successful applications of the theory is to ...
Gauges for the Ginzburg-Landau equations of superconductivity
International Nuclear Information System (INIS)
Fleckinger-Pelle, J.; Kaper, H.G.
1995-01-01
This note is concerned with gauge choices for the time-dependent Ginzburg-Landau equations of superconductivity. The requiations model the state of a superconducting sample in a magnetic field near the critical tempeature. Any two solutions related through a ''gauge transformation'' describe the same state and are physically indistinquishable. This ''gauge invariance'' can be exploited for analtyical and numerical purposes. A new gauge is proposed, which reduces the equations to a particularly attractive form
Traveling kinks in cubic nonlinear Ginzburg-Landau equations.
Rosu, H C; Cornejo-Pérez, O; Ojeda-May, P
2012-03-01
Nonlinear cubic Euler-Lagrange equations of motion in the traveling variable are usually derived from Ginzburg-Landau free energy functionals frequently encountered in several fields of physics. Many authors considered in the past damped versions of such equations, with the damping term added by hand simulating the friction due to the environment. It is known that even in this damped case kink solutions can exist. By means of a factorization method, we provide analytic formulas for several possible kink solutions of such equations of motion in the undriven and constant field driven cases, including the recently introduced Riccati parameter kinks, which were not considered previously in such a context. The latter parameter controls the delay of the switching stage of the kinks. The delay is caused by antikink components that are introduced in the structure of the solution through this parameter.
Stable solitons of quadratic ginzburg-landau equations
Crasovan; Malomed; Mihalache; Mazilu; Lederer
2000-07-01
We present a physical model based on coupled Ginzburg-Landau equations that supports stable temporal solitary-wave pulses. The system consists of two parallel-coupled cores, one having a quadratic nonlinearity, the other one being effectively linear. The former core is active, with bandwidth-limited amplification built into it, while the latter core has only losses. Parameters of the model can be easily selected so that the zero background is stable. The model has nongeneric exact analytical solutions in the form of solitary pulses ("dissipative solitons"). Direct numerical simulations, using these exact solutions as initial configurations, show that they are unstable; however, the evolution initiated by the exact unstable solitons ends up with nontrivial stable localized pulses, which are very robust attractors. Direct simulations also demonstrate that the presence of group-velocity mismatch (walkoff) between the two harmonics in the active core makes the pulses move at a constant velocity, but does not destabilize them.
On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation
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Chunyan Huang
2014-01-01
Full Text Available We study the analytic property of the (generalized quadratic derivative Ginzburg-Landau equation (1/2⩽α⩽1 in any spatial dimension n⩾1 with rough initial data. For 1/2<α⩽1, we prove the analyticity of local solutions to the (generalized quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spaces Mp,11-2α(1⩽p⩽∞. For α=1/2, we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data in B˙∞,10(ℝn∩M∞,10(ℝn. The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroup e-a+it-Δα to overcome the derivative in the nonlinear term.
Equilibrium states of a variational formulation for the Ginzburg-Landau equation
Kulikov, A. N.; Kulikov, D. A.
2017-12-01
Periodic boundary value problem for one of the versions of the complex Ginzburg- Landau equation, which is commonly called the variational Ginzburg-Landau equation are studied. Questions of existence and stability in the sense of Lyapunov, and also the local bifurcations problem of spatially nonhomogeneous equilibrium states are investigated. Three types of such solutions for the given problem are indicated. The exact formulas of the solutions for the first two types are suggested. Equilibrium states of the second type are expressed through elliptic functions. The third type of equilibrium states appears as a result of bifurcations of automodel equilibrium states, i.e., solutions of the first type in the case when the stability changes. It is shown that equilibrium states of the second and third types are unstable.
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A. A. Fonarev
2014-01-01
Full Text Available Possibility of use of a projective iterative method for search of approximations to the closed set of not trivial generalised solutions of a boundary value problem for Ginzburg - Landau's equations of the phenomenological theory of superconduction is investigated. The projective iterative method combines a projective method and iterative process. The generalised solutions of a boundary value problem for Ginzburg - Landau's equations are critical points of a functional of a superconductor free energy.
Slow time-periodic solutions of the Ginzburg-Landau equation
Doelman, A.
1989-01-01
In this paper we study the behaviour of solutions of the form if(z, t) = q~(z) e- i~wt (e << 1) of the rescaled Ginzburg-Landau equation, ~k, = [ 1 - (1 + iB)l~kl2]~k + (1 + iA)~kzz, for A = ca, B = eb, w plays the role of free parameter. This leads to a perturbation analysis on a complex Duffing
Stability of plane wave solutions in complex Ginzburg-Landau equation with delayed feedback
Puzyrev, D.; Yanchuk, S.; Vladimirov, A. G.; Gurevich, S. V.
2013-01-01
We perform bifurcation analysis of plane wave solutions in one-dimensional cubic-quintic Ginzburg-Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced. For intermediate values of the delay, bifurcation diagrams are obtained by a combination of analytical and numerical methods. For large delays, using an asymptotic approach we classify plane wave solutions into strongly unstable, weakly unstable, and ...
The Existence of Exponential Attractor for Discrete Ginzburg-Landau Equation
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Guangyin Du
2015-01-01
Full Text Available This paper studies the following discrete systems of the complex Ginzburg-Landau equation: iu˙m-(α-iε(2um-um+1-um-1+iκum+βum2σum=gm, m∈Z. Under some conditions on the parameters α, ε, κ, β, and σ, we prove the existence of exponential attractor for the semigroup associated with these discrete systems.
Facão, M.; Carvalho, M. I.
2017-10-01
In this work, we present parameter regions for the existence of stable plain solitons of the cubic complex Ginzburg-Landau equation (CGLE) with higher-order terms associated with a fourth-order expansion. Using a perturbation approach around the nonlinear Schrödinger equation soliton and a full numerical analysis that solves an ordinary differential equation for the soliton profiles and using the Evans method in the search for unstable eigenvalues, we have found that the minimum equation allowing these stable solitons is the cubic CGLE plus a term known in optics as Raman-delayed response, which is responsible for the redshift of the spectrum. The other favorable term for the occurrence of stable solitons is a term that represents the increase of nonlinear gain with higher frequencies. At the stability boundary, a bifurcation occurs giving rise to stable oscillatory solitons for higher values of the nonlinear gain. These oscillations can have very high amplitudes, with the pulse energy changing more than two orders of magnitude in a period, and they can even exhibit more complex dynamics such as period-doubling.
Blowup in the complex Ginzburg-Landau equation
Schans, Martin van der
2013-01-01
In this thesis, we study the stability of a finite-time blowup solution of a partial di erential equation (PDE). Partial di erential equations can be used to model phenomena in a wide range of applications. Examples of well known partial di erential equations are: the heat equation which models heat
Ginzburg-Landau equation and vortex liquid phase of Fermi liquid superconductors
International Nuclear Information System (INIS)
Ng, T-K; Tse, W-T
2007-01-01
In this paper we study the Ginzburg-Landau (GL) equation for Fermi liquid superconductors with strong Landau interactions F 0s and F 1s . We show that Landau interactions renormalize two parameters entering the GL equation, leading to the renormalization of the compressibility and superfluid density. The renormalization of the superfluid density in turn leads to an unconventional (2D) Berezinskii-Kosterlitz-Thouless (BKT) transition and vortex liquid phase. Application of the GL equation to describe underdoped high-T c cuprates is discussed
Ginzburg-Landau vortices driven by the Landau-Lifshitz-Gilbert equation
International Nuclear Information System (INIS)
Kurzke, Matthias; Melcher, Christof; Moser, Roger; Spirn, Daniel
2009-01-01
A simplified model for the energy of the magnetization of a thin ferromagnetic film gives rise to a version of the theory of Ginzburg-Landau vortices for sphere-valued maps. In particular we have the development of vortices as a certain parameter tends to 0. The dynamics of the magnetization is ruled by the Landau-Lifshitz-Gilbert equation, which combines characteristic properties of a nonlinear Schroedinger equation and a gradient flow. This paper studies the motion of the vortex centers under this evolution equation. (orig.)
Ginzburg-Landau vortices driven by the Landau-Lifshitz-Gilbert equation
Energy Technology Data Exchange (ETDEWEB)
Kurzke, Matthias; Melcher, Christof; Moser, Roger; Spirn, Daniel
2009-06-15
A simplified model for the energy of the magnetization of a thin ferromagnetic film gives rise to a version of the theory of Ginzburg-Landau vortices for sphere-valued maps. In particular we have the development of vortices as a certain parameter tends to 0. The dynamics of the magnetization is ruled by the Landau-Lifshitz-Gilbert equation, which combines characteristic properties of a nonlinear Schroedinger equation and a gradient flow. This paper studies the motion of the vortex centers under this evolution equation. (orig.)
Spectrum of the linearized operator for the Ginzburg-Landau equation
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Tai-Chia Lin
2000-06-01
Full Text Available We study the spectrum of the linearized operator for the Ginzburg-Landau equation about a symmetric vortex solution with degree one. We show that the smallest eigenvalue of the linearized operator has multiplicity two, and then we describe its behavior as a small parameter approaches zero. We also find a positive lower bound for all the other eigenvalues, and find estimates of the first eigenfunction. Then using these results, we give partial results on the dynamics of vortices in the nonlinear heat and Schrodinger equations.
Shokri, Ali; Afshari, Fatemeh
2015-12-01
In this article, a high-order compact alternating direction implicit (HOC-ADI) finite difference scheme is applied to numerical solution of the complex Ginzburg-Landau (GL) equation in two spatial dimensions with periodical boundary conditions. The GL equation has been used as a mathematical model for various pattern formation systems in mechanics, physics, and chemistry. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. To avoid solving the nonlinear system and to increase the accuracy and efficiency of the method, we proposed the predictor-corrector scheme. Validation of the present numerical solutions has been conducted by comparing with the exact and other methods results and evidenced a good agreement.
Statistics and Structures of Strong Turbulence in a Complex Ginzburg-Landau Equation
Iwasaki, H.; Toh, S.
1992-05-01
One-dimensional complex Ginzburg-Landau equation with a quintic nonlinearity (QCGL) is studied numerically to reveal the asymptotic property of its strong turbulence. In the inviscid limit, the QCGL equation tends to the nonlinear Schrödinger (NLS) equation which has a singular solution self-similarly blowing up in a finite time. The probability distribution function (PDF) of fluctuation amplitudes is found to have an algebraic tail with exponent close to -8. This power law is described as the multiplication of the PDF of the amplitude of a singular solution of the NLS equation and that of maximum heights of bursts. The former is shown to have a -7 power law in terms of the scaling property of the NLS singular solution. The latter is found to have a -1 power law by numerical simulation.
Data-based adjoint and H2 optimal control of the Ginzburg-Landau equation
Banks, Michael; Bodony, Daniel
2017-11-01
Equation-free, reduced-order methods of control are desirable when the governing system of interest is of very high dimension or the control is to be applied to a physical experiment. Two-phase flow optimal control problems, our target application, fit these criteria. Dynamic Mode Decomposition (DMD) is a data-driven method for model reduction that can be used to resolve the dynamics of very high dimensional systems and project the dynamics onto a smaller, more manageable basis. We evaluate the effectiveness of DMD-based forward and adjoint operator estimation when applied to H2 optimal control approaches applied to the linear and nonlinear Ginzburg-Landau equation. Perspectives on applying the data-driven adjoint to two phase flow control will be given. Office of Naval Research (ONR) as part of the Multidisciplinary University Research Initiatives (MURI) Program, under Grant Number N00014-16-1-2617.
DEFF Research Database (Denmark)
Gaididei, Yu. B.; Christiansen, Peter Leth
2008-01-01
We study a parametrically driven Ginzburg-Landau equation model with nonlinear management. The system is made of laterally coupled long active waveguides placed along a circumference. Stationary solutions of three kinds are found: periodic Ising states and two types of Bloch states, staggered and...
Mohebbi, Akbar
2018-02-01
In this paper we propose two fast and accurate numerical methods for the solution of multidimensional space fractional Ginzburg-Landau equation (FGLE). In the presented methods, to avoid solving a nonlinear system of algebraic equations and to increase the accuracy and efficiency of method, we split the complex problem into simpler sub-problems using the split-step idea. For a homogeneous FGLE, we propose a method which has fourth-order of accuracy in time component and spectral accuracy in space variable and for nonhomogeneous one, we introduce another scheme based on the Crank-Nicolson approach which has second-order of accuracy in time variable. Due to using the Fourier spectral method for fractional Laplacian operator, the resulting schemes are fully diagonal and easy to code. Numerical results are reported in terms of accuracy, computational order and CPU time to demonstrate the accuracy and efficiency of the proposed methods and to compare the results with the analytical solutions. The results show that the present methods are accurate and require low CPU time. It is illustrated that the numerical results are in good agreement with the theoretical ones.
Time-dependent Ginzburg-Landau equations for rotating and accelerating superconductors
Czech Academy of Sciences Publication Activity Database
Lipavský, P.; Bok, J.; Koláček, Jan
2013-01-01
Roč. 492, Sept (2013), 144-151 ISSN 0921-4534 R&D Projects: GA ČR(CZ) GAP204/11/0015 Institutional support: RVO:68378271 Keywords : superconductivity * Ginzburg-Landau theory * London field Subject RIV: BM - Solid Matter Physics ; Magnetism Impact factor: 1.110, year: 2013
International Nuclear Information System (INIS)
Deissler, R.J.; Brand, H.R.; Deissler, R.J.; Brand, H.R.
1998-01-01
We study the effect of nonlinear gradient terms on breathing localized solutions in the complex Ginzburg-Landau equation. It is found that even small nonlinear gradient terms emdash which appear at the same order as the quintic term emdash can cause dramatic changes in the behavior of the solution, such as causing opposite sides of an otherwise monoperiodic symmetrically breathing solution to breathe at different frequencies, thus causing the solution to breathe periodically or chaotically on only one side or the solution to rapidly spread. copyright 1998 The American Physical Society
Construction of a Blow-Up Solution for the Complex Ginzburg-Landau Equation in a Critical Case
Nouaili, Nejla; Zaag, Hatem
2017-12-01
We construct a solution for the Complex Ginzburg-Landau equation in a critical case which blows up in finite time T only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows us to prove the stability of the constructed solution.
International Nuclear Information System (INIS)
Fewo, Serge I.; Kofane, Timoleon C.; Ngabireng, Claude M.
2008-01-01
With the help of the Maxwell equations, a basic equation modeling the propagation of ultrashort optical solitons in optical fiber is derived, namely the higher-order complex Ginzburg-Landau equation (HCGLE). Considering this one-dimensional HCGLE, we obtain a set of differential equations characterizing the variation of the pulse parameters called collective variables (CVs), of a pulse propagating in dispersion-managed (DM) fiber optic-links. Equations obtained are investigated numerically in order to observe the behaviour of pulse parameters along the optical fiber. A fully numerical simulation of the one-dimensional HCGLE finally tests the results of the CV theory. A good agreement between both methods is observed. Among various behaviours, chaotic pulses, attenuate pulses and stable pulses can be obtained under certain parameter values. (author)
Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains II: The monotone case
Zhou, Feng; Sun, Chunyou; Cheng, Jiaqi
2018-02-01
In this article, we continue the study of the dynamics of the following complex Ginzburg-Landau equation ∂tu - (λ + iα)Δu + (κ + iβ)|u|p-2u - γu = f(t) on non-cylindrical domains. We assume that the spatial domains are bounded and increase with time, which is different from the diffeomorphism case presented in Zhou and Sun [Discrete Contin. Dyn. Syst., Ser. B 21, 3767-3792 (2016)]. We develop a new penalty function to establish the existence and uniqueness of a variational solution satisfying energy equality as well as some energy inequalities and prove the existence of a D -pullback attractor for the non-autonomous dynamical system generated by this class of solutions.
Gelens, Lendert; Matías, Manuel A; Gomila, Damià; Dorissen, Tom; Colet, Pere
2014-01-01
We study the influence of a linear nonlocal spatial coupling on the interaction of fronts connecting two equivalent stable states in the prototypical 1-dimensional real Ginzburg-Landau equation. While for local coupling the fronts are always monotonic and therefore the dynamical behavior leads to coarsening and the annihilation of pairs of fronts, nonlocal terms can induce spatial oscillations in the front, allowing for the creation of localized structures, emerging from pinning between two fronts. We show this for three different nonlocal influence kernels. The first two, mod-exponential and Gaussian, are positive definite and decay exponentially or faster, while the third one, a Mexican-hat kernel, is not positive definite.
Bethuel, Fabrice; Helein, Frederic
2017-01-01
This book is concerned with the study in two dimensions of stationary solutions of uɛ of a complex valued Ginzburg-Landau equation involving a small parameter ɛ. Such problems are related to questions occurring in physics, e.g., phase transition phenomena in superconductors and superfluids. The parameter ɛ has a dimension of a length which is usually small. Thus, it is of great interest to study the asymptotics as ɛ tends to zero. One of the main results asserts that the limit u-star of minimizers uɛ exists. Moreover, u-star is smooth except at a finite number of points called defects or vortices in physics. The number of these defects is exactly the Brouwer degree – or winding number – of the boundary condition. Each singularity has degree one – or as physicists would say, vortices are quantized. The singularities have infinite energy, but after removing the core energy we are lead to a concept of finite renormalized energy. The location of the singularities is completely determined by minimiz...
Numerical Analysis of Ginzburg-Landau Models for Superconductivity.
Coskun, Erhan
Thin film conventional, as well as High T _{c} superconductors of various geometric shapes placed under both uniform and variable strength magnetic field are studied using the universially accepted macroscopic Ginzburg-Landau model. A series of new theoretical results concerning the properties of solution is presented using the semi -discrete time-dependent Ginzburg-Landau equations, staggered grid setup and natural boundary conditions. Efficient serial algorithms including a novel adaptive algorithm is developed and successfully implemented for solving the governing highly nonlinear parabolic system of equations. Refinement technique used in the adaptive algorithm is based on modified forward Euler method which was also developed by us to ease the restriction on time step size for stability considerations. Stability and convergence properties of forward and modified forward Euler schemes are studied. Numerical simulations of various recent physical experiments of technological importance such as vortes motion and pinning are performed. The numerical code for solving time-dependent Ginzburg-Landau equations is parallelized using BlockComm -Chameleon and PCN. The parallel code was run on the distributed memory multiprocessors intel iPSC/860, IBM-SP1 and cluster of Sun Sparc workstations, all located at Mathematics and Computer Science Division, Argonne National Laboratory.
A non-existence result for the Ginzburg-Landau equations
DEFF Research Database (Denmark)
Kachmar, Ayman; Persson, Mikael
2009-01-01
We consider the stationary Ginzburg–Landau equations in , d=2,3 . We exhibit a class of applied magnetic fields (including constant fields) such that the Ginzburg–Landau equations do not admit finite energy solutions....
International Nuclear Information System (INIS)
Fewo, S I; Kenfack-Jiotsa, A; Kofane, T C
2006-01-01
With the help of the one-dimensional quintic complex Ginzburg-Landau equation (CGLE) as perturbations of the nonlinear Schroedinger equation (NLSE), we derive the equations of motion of pulse parameters called collective variables (CVs), of a pulse propagating in dispersion-managed (DM) fibre optic links. The equations obtained are investigated numerically in order to view the evolution of pulse parameters along the propagation distance, and also to analyse effects of initial amplitude and width on the propagating pulse. Nonlinear gain is shown to be beneficial in stabilizing DM solitons. A fully numerical simulation of the one-dimensional quintic CGLE as perturbations of NLSE finally tests the results of the CV theory. A good agreement is observed between both methods
Boundary conditions in the Ginzburg Landau Formulation in heavy ...
African Journals Online (AJOL)
The linearized gap equation is the basis for the microscopic derivation of the second order terms in the Ginzburg-Landau free energy expansion. However, close to the boundary these second order terms do not have the same form, since the kernel is changed due to quasi-particle scattering. In addition, these boundary ...
Ginzburg-Landau-type theory of nonpolarized spin superconductivity
Lv, Peng; Bao, Zhi-qiang; Guo, Ai-Min; Xie, X. C.; Sun, Qing-Feng
2017-01-01
Since the concept of spin superconductor was proposed, all the related studies concentrate on the spin-polarized case. Here, we generalize the study to the spin-non-polarized case. The free energy of nonpolarized spin superconductor is obtained, and Ginzburg-Landau-type equations are derived by using the variational method. These Ginzburg-Landau-type equations can be reduced to the spin-polarized case when the spin direction is fixed. Moreover, the expressions of super linear and angular spin currents inside the superconductor are derived. We demonstrate that the electric field induced by the super spin current is equal to the one induced by an equivalent charge obtained from the second Ginzburg-Landau-type equation, which shows self-consistency of our theory. By applying these Ginzburg-Landau-type equations, the effect of electric field on the superconductor is also studied. These results will help us get a better understanding of the spin superconductor and related topics such as the Bose-Einstein condensate of magnons and spin superfluidity.
Morales, M A; Rojas, J F; Oliveros, J; Hernández S, A A
2015-03-07
In this work the skin coating of some vertebrate marine animals is modeled considering only dermis, epidermis and basal layers. The biological process takes into account: cellular diffusion of the epidermis, diffusion inhibition and long-range spatial interaction (nonlocal effect on diffusive dispersal) for cells of dermal tissue. The chemical and physical interactions between dermis and epidermis are represented by coupling quadratic terms and nonlinear terms additional. The model presents an interesting property associated with their gradient form: a connection between some physical, chemical and biological systems. The model equations proposed are solved with numerical methods to study the spatially stable emergent configurations. The spatiotemporal dynamic obtained of the numerical solution of these equations, present similarity with biological behaviors that have been found recently in the cellular movement of chromatophores (as contact-dependent depolarization and repulsion movement between melanophores, xanthophores and iridophores). The numerical solution of the model shows a great variety of beautiful patterns that are robust to changes of boundary condition. The resultant patterns are very similar to the pigmentation of some fish. Copyright © 2014 Elsevier Ltd. All rights reserved.
Efficient solution of 3D Ginzburg-Landau problem for mesoscopic superconductors
Pereira, Paulo J.; Moshchalkov, Victor V.; Chibotaru, Liviu F.
2014-03-01
The recently proposed approach for the solution of Ginzburg-Landau (GL) problem for 2D samples of arbitrary shape is, in this article, extended over 3D samples having the shape of (i) a prism with arbitrary base and (ii) a solid of revolution with arbitrary profile. Starting from the set of Laplace operator eigenfunctions of a 2D object, we construct an approximation to or the exact eigenfunctions of the Laplace operator of a 3D structure by applying an extrusion or revolution to these solutions. This set of functions is used as the basis to construct the solutions of the linearized GL equation. These solutions are then used as basis for the non-linear GL equation much like the famous LCAO method. To solve the non-linear equation, we used the Newton-Raphson method starting from the solution of the linear equation, i.e., the nucleation distribution of superconducting condensate. The vector potential approximations typically used in 2D cases, i.e., considering it as corresponding to applied constant field, are in the 3D case harder to justify. For that reason, we use a locally corrected Nystrom method to solve the second Ginzburg-Landau equation. The complete solution of GL problem is then achieved by solving self-consistently both equations.
Microscopic Derivation of Ginzburg-Landau Theory
DEFF Research Database (Denmark)
Frank, Rupert; Hainzl, Christian; Seiringer, Robert
2012-01-01
We give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical...
Self-dual Ginzburg-Landau vortices in a disc
Energy Technology Data Exchange (ETDEWEB)
Lozano, G.S. [Departamento de Fisica, FCEyN, Universidad de Buenos Aires, Pab I, Ciudad Univeristaria, Buenos Aires (Argentina); Consejo Nacional de Investigaciones Cientificas y Tecnicas (Argentina); Manias, M.V.; Moreno, E.F. [Consejo Nacional de Investigaciones Cientificas y Tecnicas (Argentina); Departamento de Fisica, Universidad Nacional de La Plata, La Plata (Argentina)
2001-07-20
We study the properties of the Ginzburg-Landau model in the self-dual point for a two-dimensional finite system. By a numerical calculation we analyse the solutions of the Euler-Lagrange equations for a cylindrically symmetric ansatz. We also study the self-dual equations for this case. We find that the minimal energy configurations are not given by the Bogomol'nyi equations but by solutions to the Euler-Lagrange ones. With a simple approximation scheme we reproduce the result of the numerical calculation. (author)
Domain Walls and Textured Vortices in a Two-Component Ginzburg-Landau Model
DEFF Research Database (Denmark)
Madsen, Søren Peder; Gaididei, Yu. B.; Christiansen, Peter Leth
2005-01-01
We look for domain wall and textured vortex solutions in a two-component Ginzburg-Landau model inspired by two-band superconductivity. The two-dimensional two-component model, with equal coherence lengths and no magnetic field, shows some interesting properties. In the absence of a Josephson type...... coupling between the two order parameters a ''textured vortex'' is found by analytical and numerical solution of the Ginzburg-Landau equations. With a Josephson type coupling between the two order parameters we find the system to split up in two domains separated by a domain wall, where the order parameter...
Ginzburg-Landau approach to color superconductivity
International Nuclear Information System (INIS)
Iida, Kei; Baym, Gordon; Matsuura, Taeko; Hatsuda, Tetsuo
2004-01-01
We study, within Ginzburg-Landau theory, the equilibrium properties of a color superconductor of three-flavor massless quarks near the transition temperature. We first draw the phase diagram in the space of the parameters controlling the thermodynamic-potential terms of fourth order in the pairing gap. Within the color and flavor antitriplet channel with zero total angular momentum, the phase diagram contains the color-flavor locked and two-flavor pairing phases; the limit of weak coupling is included in the color-flavor locked phase. The responses of the color-flavor locked and two-flavor condensates to external magnetic fields and rotation are then investigated by calculating the induced supercurrents. In equilibrium, rotation can produce vortices only in the color-flavor locked condensate. We finally discuss the effects of fluctuations in gauge fields and order parameters on the finite temperature phase transition. (author)
Microscopic Derivation of the Ginzburg-Landau Model
DEFF Research Database (Denmark)
Frank, Rupert; Hainzl, Christian; Seiringer, Robert
2014-01-01
We present a summary of our recent rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit...
On the Ginzburg-Landau critical field in three dimensions
DEFF Research Database (Denmark)
Fournais, Søren; Helffer, Bernard
2009-01-01
We study the three-dimensional Ginzburg-Landau model of superconductivity. Several natural definitions of the (third) critical field, HC3, governing the transition from the superconducting state to the normal state, are considered. We analyze the relation between these fields and give conditions ...... of the field strength for sufficiently large fields. As a consequence of our analysis, we give an affirmative answer to a conjecture by Pan. © 2008 Wiley Periodicals, Inc....
Boundary condition for Ginzburg-Landau theory of superconducting layers
Czech Academy of Sciences Publication Activity Database
Koláček, Jan; Lipavský, Pavel; Morawetz, K.; Brandt, E. H.
2009-01-01
Roč. 79, č. 17 (2009), 174510/1-174510/6 ISSN 1098-0121 R&D Projects: GA ČR GA202/08/0326; GA AV ČR IAA100100712 Institutional research plan: CEZ:AV0Z10100521 Keywords : superconductivity * Ginzburg-Landau theory Subject RIV: BM - Solid Matter Physics ; Magnetism Impact factor: 3.475, year: 2009
Boundary value problem in the theory of Ginzburg-Landau
Energy Technology Data Exchange (ETDEWEB)
Boutet de Monvel-Berthier, A.M.; Georgescu, V.; Purice, R.
1988-06-01
We study an elliptic problem related to the Ginzburg-Landau model for the supraconductivity. We reduce the problem to a two-dimensional problem with an infinite dimensional symmetry group. We define the topological degree of a function of class H/sup 1/2/ and modulus one, defined on a plane curve diffeomorphic to a circle. We study the topological structure of the configuration space.
Carty, G. J.; Hampshire, D. P.
2008-01-01
In polycrystalline superconducting materials optimized for high critical current density (JC) in high magnetic fields, the mechanism that determines JC has long remained uncertain because of the complicated manner in which the fluxon-fluxon and fluxon-microstructure forces combine. In this work, the time-dependent Ginzburg-Landau equations are used to produce visualizations of fluxons at JC that show the disorder in the pinned part of the flux-line lattice and the motion of those fluxons alon...
Di Patti, Francesca; Fanelli, Duccio; Miele, Filippo; Carletti, Timoteo
2018-03-01
A normal form approximation for the evolution of a reaction-diffusion system hosted on a directed graph is derived, in the vicinity of a supercritical Hopf bifurcation. Weak diffusive couplings are assumed to hold between adjacent nodes. Under this working assumption, a Complex Ginzburg-Landau equation (CGLE) is obtained, whose coefficients depend on the parameters of the model and the topological characteristics of the underlying network. The CGLE enables one to probe the stability of the synchronous oscillating solution, as displayed by the reaction-diffusion system above Hopf bifurcation. More specifically, conditions can be worked out for the onset of the symmetry breaking instability that eventually destroys the uniform oscillatory state. Numerical tests performed for the Brusselator model confirm the validity of the proposed theoretical scheme. Patterns recorded for the CGLE resemble closely those recovered upon integration of the original Brussellator dynamics.
An Approach to Quad Meshing Based On Cross Valued Maps and the Ginzburg-Landau Theory
Energy Technology Data Exchange (ETDEWEB)
Viertel, Ryan [Univ. of Utah, Salt Lake City, UT (United States). Dept. of Mathematics; Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Osting, Braxton [Univ. of Utah, Salt Lake City, UT (United States). Dept. of Mathematics
2017-08-01
A generalization of vector fields, referred to as N-direction fields or cross fields when N=4, has been recently introduced and studied for geometry processing, with applications in quadrilateral (quad) meshing, texture mapping, and parameterization. We make the observation that cross field design for two-dimensional quad meshing is related to the well-known Ginzburg-Landau problem from mathematical physics. This identification yields a variety of theoretical tools for efficiently computing boundary-aligned quad meshes, with provable guarantees on the resulting mesh, for example, the number of mesh defects and bounds on the defect locations. The procedure for generating the quad mesh is to (i) find a complex-valued "representation" field that minimizes the Dirichlet energy subject to a boundary constraint, (ii) convert the representation field into a boundary-aligned, smooth cross field, (iii) use separatrices of the cross field to partition the domain into four sided regions, and (iv) mesh each of these four-sided regions using standard techniques. Under certain assumptions on the geometry of the domain, we prove that this procedure can be used to produce a cross field whose separatrices partition the domain into four sided regions. To solve the energy minimization problem for the representation field, we use an extension of the Merriman-Bence-Osher (MBO) threshold dynamics method, originally conceived as an algorithm to simulate motion by mean curvature, to minimize the Ginzburg-Landau energy for the optimal representation field. Lastly, we demonstrate the method on a variety of test domains.
Carty, George J.; Hampshire, Damian P.
2008-05-01
In polycrystalline superconducting materials optimized for high critical current density (JC) in high magnetic fields, the mechanism that determines JC has long remained uncertain because of the complicated manner in which the fluxon-fluxon and fluxon-microstructure forces combine. In this work, the time-dependent Ginzburg-Landau equations are used to produce visualizations of fluxons at JC that show the disorder in the pinned part of the flux-line lattice and the motion of those fluxons along grain boundaries that cause dissipation. Calculated values of JC are consistent with experimental data.
Theory of a condensed charged-Bose, charged Fermi gas and Ginzburg--Landau studies of superfluid 3He
International Nuclear Information System (INIS)
Dahl, D.A.
1976-01-01
Two independent topics in the field of condensed matter physics are examined: the condensed charged-Bose, charged Fermi gas and superfluid 3 He. Green's function (field theoretic) methods are used to derive the low-temperature properties of a dense, neutral gas of condensed charged bosons and degenerate charged fermions. Restriction is made to the case where the fermion mass is much lighter than the boson mass. Linear response and the density-density correlation function are examined and shown to exhibit two collective modes: a plasmon branch and a phonon branch with speed equal to that of ionic sound in solids. Comparison with a possible astrophysical application (white dwarf stars) is made. The behavior near the superfluid transition temperature (Ginzburg--Landau regime) of 3 He is then studied. Gorkov equations are derived and studied in the weak-coupling limit. In this way the form and order of magnitude estimates of coefficients appearing in the Ginzburg--Landau theory are obtained. Weak-coupling particle and spin currents are derived. Various perturbations break the large degeneracy of the states and have experimental implications. The electric contribution to the Ginzburg--Landau free energy is studied for the proposed A and B phases. Imposition of an electric field orients the axial state, but does not give rise to shifts in the NMR resonances. Shifts and discontinuous jumps in the longitudinal and transverse signals are predicted for the Balian--Werthamer state, the details depending on the relative strengths of the fields, as well as the angle between them
Merkurjev, Ekaterina; Bertozzi, Andrea; Yan, Xiaoran; Lerman, Kristina
2017-07-01
Recent advances in clustering have included continuous relaxations of the Cheeger cut problem and those which address its linear approximation using the graph Laplacian. In this paper, we show how to use the graph Laplacian to solve the fully nonlinear Cheeger cut problem, as well as the ratio cut optimization task. Both problems are connected to total variation minimization, and the related Ginzburg-Landau functional is used in the derivation of the methods. The graph framework discussed in this paper is undirected. The resulting algorithms are efficient ways to cluster the data into two classes, and they can be easily extended to the case of multiple classes, or used on a multiclass data set via recursive bipartitioning. In addition to showing results on benchmark data sets, we also show an application of the algorithm to hyperspectral video data.
Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint
DEFF Research Database (Denmark)
Kachmar, Ayman
2010-01-01
This paper is devoted to an analysis of vortex-nucleation for a Ginzburg-Landau functional with discontinuous constraint. This functional has been proposed as a model for vortex-pinning, and usually accounts for the energy resulting from the interface of two superconductors. The critical applied ...
Derivation of Ginzburg-Landau theory for a one-dimensional system with contact interaction
DEFF Research Database (Denmark)
Frank, Rupert; Hanizl, Christian; Seiringer, Robert
2013-01-01
In a recent paper we give the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Here we present our results in the simplified case of a one-dimensional system of particles interacting via a delta-potential....
Czech Academy of Sciences Publication Activity Database
Lin, P.-J.; Lipavský, Pavel
2008-01-01
Roč. 77, č. 14 (2008), 144505/1-144505/16 ISSN 1098-0121 Institutional research plan: CEZ:AV0Z10100521 Keywords : non-equilibrium superconductivity * Ginzburg-Landau theory Subject RIV: BE - Theoretical Physics Impact factor: 3.322, year: 2008
Belmiloudi, Aziz
2006-01-01
We formulate and study robust control problems for a two-dimensional time-dependent Ginzburg-Landau model with Robin boundary conditions on phase-field parameter, which describes the phase transitions taking place in superconductor films with variable thickness. The objective of such study is to control the motion of vortices in the superconductor films by taking into account the influence of noises in data. Firstly, we introduce the perturbation problem of the nonlinear ...
Critical behaviour of the Ginzburg-Landau model in the type II region
Kajantie, K.; Neuhaus, T.; Rajantie, A.; Rummukainen, K.
2002-01-01
We study the critical behaviour of the three-dimensional U(1) gauge+Higgs theory (Ginzburg-Landau model) at large scalar self-coupling \\lambda (``type II region'') by measuring various correlation lengths as well as the Abrikosov-Nielsen-Olesen vortex tension. We identify different scaling regions as the transition is approached from below, and carry out detailed comparisons with the criticality of the 3d O(2) symmetric scalar theory.
Electrostatic field in superconductors IV: theory of Ginzburg-Landau type
Czech Academy of Sciences Publication Activity Database
Lipavský, Pavel; Koláček, Jan
2009-01-01
Roč. 23, 20-21 (2009), s. 4505-4511 ISSN 0217-9792 R&D Projects: GA ČR GA202/04/0585; GA ČR GA202/05/0173; GA AV ČR IAA1010312 Institutional research plan: CEZ:AV0Z10100521 Keywords : superconductivity * Ginzburg-Landau theory Subject RIV: BM - Solid Matter Physics ; Magnetism Impact factor: 0.408, year: 2009
Hc2 of anisotropy two-band superconductors by Ginzburg-Landau approach
International Nuclear Information System (INIS)
Udomsamuthirun, P.; Changjan, A.; Kumvongsa, C.; Yoksan, S.
2006-01-01
The purpose of this research is to study the upper critical field H c2 of two-band superconductors by two-band Ginzburg-Landau approach. The analytical formula of H c2 included anisotropy of order parameter and anisotropy of effective-mass are found. The parameters of the upper critical field in ab-plane (H c2 - bar ab ) and c-axis (H c2 - bar c ) can be found by fitting to the experimental data. Finally, we can find the ratio of upper critical field that temperature dependent in the range of experimental result
Yun, Gunsu; Oh, Youngmin; Lee, Jieun; Hwang, H. J.; Lee, Jaehyun; Leconte, Michael; Kstar Team
2017-10-01
The boundary of high-temperature plasma confined by a toroidal magnetic field structure often undergoes quasi-periodic relaxation oscillations between high and low energy states. On the KSTAR tokamak, the oscillation cycle consists of a long quasi-steady state characterized by eigenmode-like filamentary modes, an abrupt transition into non-modal filamentary structure [Lee JE, Sci. Rep. 7, 45075], and its rapid burst (via magnetic reconnection) leading to the boundary collapse. A phenomenological model including the effects of time-varying perpendicular flow shear, turbulent transport, and external heating has been developed to understand the nonlinear oscillation. The model, which has the form of a generalized complex Ginzburg-Landau equation, shows that the flow shear amplitude and the shear layer width determine the nonlinear oscillation. Numerical solutions revealed that there exists a critical flow shear level below which steady states can exist. This result suggests that the abrupt transition to the non-modal unstable state is due to the flow shear increasing above the critical level. The model predicts that high wavenumber (k) modes can coexist with low- k modes at sufficiently low level of flow shear [Lee J, Phys. Rev. Lett. 117, 075001]. Work supported by the National Research Foundation of Korea.
Coarse graining from variationally enhanced sampling applied to the Ginzburg-Landau model
Invernizzi, Michele; Valsson, Omar; Parrinello, Michele
2017-03-01
A powerful way to deal with a complex system is to build a coarse-grained model capable of catching its main physical features, while being computationally affordable. Inevitably, such coarse-grained models introduce a set of phenomenological parameters, which are often not easily deducible from the underlying atomistic system. We present a unique approach to the calculation of these parameters, based on the recently introduced variationally enhanced sampling method. It allows us to obtain the parameters from atomistic simulations, providing thus a direct connection between the microscopic and the mesoscopic scale. The coarse-grained model we consider is that of Ginzburg-Landau, valid around a second-order critical point. In particular, we use it to describe a Lennard-Jones fluid in the region close to the liquid-vapor critical point. The procedure is general and can be adapted to other coarse-grained models.
Trade-offs for feedback control of the linearized Ginzburg-Landau system
Illingworth, Simon; Oehler, Stephan
2017-11-01
We consider feedback control of the linearized Ginzburg-Landau system. The particular focus is on any trade-offs present in the single-input single-output control problem. The work is in three parts. First, we consider the estimation problem in which a single sensor is used to estimate the entire flow field (without any control). By considering the optimal sensor placement with varying system stability, a fundamental trade-off for the estimation problem is made clear. Second, we consider the full-information control problem in which the entire flow field is known, but only a single actuator is available for control. We show that a similar trade-off exists when placing the single actuator. Third, we consider the overall feedback control problem in which only a single sensor is available for measurement; and only a single actuator is available for control. By varying the system stability, a similar fundamental trade-off is made clear. Implications for effective feedback control with a single sensor and a single actuator are discussed.
Ginzburg-Landau expansion in strongly disordered attractive Anderson-Hubbard model
Kuchinskii, E. Z.; Kuleeva, N. A.; Sadovskii, M. V.
2017-07-01
We have studied disordering effects on the coefficients of Ginzburg-Landau expansion in powers of superconducting order parameter in the attractive Anderson-Hubbard model within the generalized DMFT+Σ approximation. We consider the wide region of attractive potentials U from the weak coupling region, where superconductivity is described by BCS model, to the strong coupling region, where the superconducting transition is related with Bose-Einstein condensation (BEC) of compact Cooper pairs formed at temperatures essentially larger than the temperature of superconducting transition, and a wide range of disorder—from weak to strong, where the system is in the vicinity of Anderson transition. In the case of semielliptic bare density of states, disorder's influence upon the coefficients A and B of the square and the fourth power of the order parameter is universal for any value of electron correlation and is related only to the general disorder widening of the bare band (generalized Anderson theorem). Such universality is absent for the gradient term expansion coefficient C. In the usual theory of "dirty" superconductors, the C coefficient drops with the growth of disorder. In the limit of strong disorder in BCS limit, the coefficient C is very sensitive to the effects of Anderson localization, which lead to its further drop with disorder growth up to the region of the Anderson insulator. In the region of BCS-BEC crossover and in BEC limit, the coefficient C and all related physical properties are weakly dependent on disorder. In particular, this leads to relatively weak disorder dependence of both penetration depth and coherence lengths, as well as of related slope of the upper critical magnetic field at superconducting transition, in the region of very strong coupling.
Non-radially symmetric solutions to the Ginzburg-Landau equation
Ovchinnikov, Yu N
2000-01-01
We study an atom with finitely many energy levels in contact with a heat bath consisting of photons (black body radiation) at a temperature $T >0$. The dynamics of this system is described by a Liouville operator, or thermal Hamiltonian, which is the sum of an atomic Liouville operator, of a Liouville operator describing the dynamics of a free, massless Bose field, and a local operator describing the interactions between the atom and the heat bath. We show that an arbitrary initial state which is normal with respect to the equilibrium state of the uncoupled system at temperature $T$ converges to an equilibrium state of the coupled system at the same temperature, as time tends to $+ \\infty$
Solving Equations Applet Project
Thatcher, Kimberly
2011-01-01
The purpose of this paper is to summarize a Masters Project for the MMath Degree. The purpose of the project was to create and evaluate an applet that maintains the advantages of the existent manipulatives (Hands-On Equations® and the NLVM applet) while also overcoming the limitations of each. Another product of this project is accompanying lesson plans for teachers.
Solving Differential Equations in R: Package deSolve
Soetaert, K.E.R.; Petzoldt, T.; Setzer, R.W.
2010-01-01
In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines approach. The
Golubov, Alexandre Avraamovitch; Koshelev, A.E.
2003-01-01
We investigate the upper critical field in a dirty two-band superconductor within quasiclassical Usadel equations. The regime of very high anisotropy in the quasi-2D band, relevant for MgB2, is considered. We show that strong disparities in pairing interactions and diffusion constant anisotropies
A Versatile Technique for Solving Quintic Equations
Kulkarni, Raghavendra G.
2006-01-01
In this paper we present a versatile technique to solve several types of solvable quintic equations. In the technique described here, the given quintic is first converted to a sextic equation by adding a root, and the resulting sextic equation is decomposed into two cubic polynomials as factors in a novel fashion. The resultant cubic equations are…
Differential equations inverse and direct problems
Favini, Angelo
2006-01-01
DEGENERATE FIRST ORDER IDENTIFICATION PROBLEMS IN BANACH SPACES A NONISOTHERMAL DYNAMICAL GINZBURG-LANDAU MODEL OF SUPERCONDUCTIVITY. EXISTENCE AND UNIQUENESS THEOREMSSOME GLOBAL IN TIME RESULTS FOR INTEGRODIFFERENTIAL PARABOLIC INVERSE PROBLEMSFOURTH ORDER ORDINARY DIFFERENTIAL OPERATORS WITH GENERAL WENTZELL BOUNDARY CONDITIONSTUDY OF ELLIPTIC DIFFERENTIAL EQUATIONS IN UMD SPACESDEGENERATE INTEGRODIFFERENTIAL EQUATIONS OF PARABOLIC TYPE EXPONENTIAL ATTRACTORS FOR SEMICONDUCTOR EQUATIONSCONVERGENCE TO STATIONARY STATES OF SOLUTIONS TO THE SEMILINEAR EQUATION OF VISCOELASTICITY ASYMPTOTIC BEHA
Solving Cubic Equations by Polynomial Decomposition
Kulkarni, Raghavendra G.
2011-01-01
Several mathematicians struggled to solve cubic equations, and in 1515 Scipione del Ferro reportedly solved the cubic while participating in a local mathematical contest, but did not bother to publish his method. Then it was Cardano (1539) who first published the solution to the general cubic equation in his book "The Great Art, or, The Rules of…
Solving equations by topological methods
Directory of Open Access Journals (Sweden)
Lech Górniewicz
2005-01-01
Full Text Available In this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in differential equations.
Managing Element Interactivity in Equation Solving
Ngu, Bing Hiong; Phan, Huy P.; Yeung, Alexander Seeshing; Chung, Siu Fung
2018-01-01
Between two popular teaching methods (i.e., balance method vs. inverse method) for equation solving, the main difference occurs at the operational line (e.g., +2 on both sides vs. -2 becomes +2), whereby it alters the state of the equation and yet maintains its equality. Element interactivity occurs on both sides of the equation in the balance…
Deterministic equation solving over finite fields
Woestijne, Christiaan Evert van de
2006-01-01
It is shown how to solve diagonal forms in many variables over finite fields by means of a deterministic efficient algorithm. Applications to norm equations, quadratic forms, and elliptic curves are given.
Solving Differential Equations in R: Package deSolve
Directory of Open Access Journals (Sweden)
Karline Soetaert
2010-02-01
Full Text Available In this paper we present the R package deSolve to solve initial value problems (IVP written as ordinary differential equations (ODE, differential algebraic equations (DAE of index 0 or 1 and partial differential equations (PDE, the latter solved using the method of lines approach. The differential equations can be represented in R code or as compiled code. In the latter case, R is used as a tool to trigger the integration and post-process the results, which facilitates model development and application, whilst the compiled code significantly increases simulation speed. The methods implemented are efficient, robust, and well documented public-domain Fortran routines. They include four integrators from the ODEPACK package (LSODE, LSODES, LSODA, LSODAR, DVODE and DASPK2.0. In addition, a suite of Runge-Kutta integrators and special-purpose solvers to efficiently integrate 1-, 2- and 3-dimensional partial differential equations are available. The routines solve both stiff and non-stiff systems, and include many options, e.g., to deal in an efficient way with the sparsity of the Jacobian matrix, or finding the root of equations. In this article, our objectives are threefold: (1 to demonstrate the potential of using R for dynamic modeling, (2 to highlight typical uses of the different methods implemented and (3 to compare the performance of models specified in R code and in compiled code for a number of test cases. These comparisons demonstrate that, if the use of loops is avoided, R code can efficiently integrate problems comprising several thousands of state variables. Nevertheless, the same problem may be solved from 2 to more than 50 times faster by using compiled code compared to an implementation using only R code. Still, amongst the benefits of R are a more flexible and interactive implementation, better readability of the code, and access to R’s high-level procedures. deSolve is the successor of package odesolve which will be deprecated in
Students’ difficulties in solving linear equation problems
Wati, S.; Fitriana, L.; Mardiyana
2018-03-01
A linear equation is an algebra material that exists in junior high school to university. It is a very important material for students in order to learn more advanced mathematics topics. Therefore, linear equation material is essential to be mastered. However, the result of 2016 national examination in Indonesia showed that students’ achievement in solving linear equation problem was low. This fact became a background to investigate students’ difficulties in solving linear equation problems. This study used qualitative descriptive method. An individual written test on linear equation tasks was administered, followed by interviews. Twenty-one sample students of grade VIII of SMPIT Insan Kamil Karanganyar did the written test, and 6 of them were interviewed afterward. The result showed that students with high mathematics achievement donot have difficulties, students with medium mathematics achievement have factual difficulties, and students with low mathematics achievement have factual, conceptual, operational, and principle difficulties. Based on the result there is a need of meaningfulness teaching strategy to help students to overcome difficulties in solving linear equation problems.
The Use of Transformations in Solving Equations
Libeskind, Shlomo
2010-01-01
Many workshops and meetings with the US high school mathematics teachers revealed a lack of familiarity with the use of transformations in solving equations and problems related to the roots of polynomials. This note describes two transformational approaches to the derivation of the quadratic formula as well as transformational approaches to…
A method for solving neutron transport equation
International Nuclear Information System (INIS)
Dimitrijevic, Z.
1993-01-01
The procedure for solving the transport equation by directly integrating for case one-dimensional uniform multigroup medium is shown. The solution is expressed in terms of linear combination of function H n (x,μ), and the coefficient is determined from given conditions. The solution is applied for homogeneous slab of critical thickness. (author)
Adaptive wavelet algorithms for solving operator equations
Gantumur, T.
2006-01-01
This thesis treats various aspects of adaptive wavelet algorithms for solving operator equations. For a separable Hilbert space H, a linear functional f in H', and a boundedly invertible linear operator A:H->H', we consider the problem of finding u from H satisfying Au=f. Typically A is given by a
A New time Integration Scheme for Cahn-hilliard Equations
Schaefer, R.
2015-06-01
In this paper we present a new integration scheme that can be applied to solving difficult non-stationary non-linear problems. It is obtained by a successive linearization of the Crank- Nicolson scheme, that is unconditionally stable, but requires solving non-linear equation at each time step. We applied our linearized scheme for the time integration of the challenging Cahn-Hilliard equation, modeling the phase separation in fluids. At each time step the resulting variational equation is solved using higher-order isogeometric finite element method, with B- spline basis functions. The method was implemented in the PETIGA framework interfaced via the PETSc toolkit. The GMRES iterative solver was utilized for the solution of a resulting linear system at every time step. We also apply a simple adaptivity rule, which increases the time step size when the number of GMRES iterations is lower than 30. We compared our method with a non-linear, two stage predictor-multicorrector scheme, utilizing a sophisticated step length adaptivity. We controlled the stability of our simulations by monitoring the Ginzburg-Landau free energy functional. The proposed integration scheme outperforms the two-stage competitor in terms of the execution time, at the same time having a similar evolution of the free energy functional.
Solving Nonlinear Euler Equations with Arbitrary Accuracy
Dyson, Rodger W.
2005-01-01
A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.
Spectral difference methods for solving differential equations
Mazziotti, David A.
1999-01-01
A family of recently developed techniques is explored for achieving both matrix sparsity and rapid convergence when numerically solving differential and eigenvalue equations without domain decomposition. These methods, which we call spectral differences, include Boyd's sum acceleration techniques and the Lagrange distributed approximating functional (LDAF) approach. A formula is developed for estimating the unknown Gaussian parameter within LDAF. We implement these methods to calculate the Morse vibrational energies for diatomic iodine. For equivalent bandwidths the sum acceleration with finite difference weights generates energies which are between two and three orders of magnitude more accurate than those from LDAF.
DEFF Research Database (Denmark)
Milovanov, A.V.; Juul Rasmussen, J.
2005-01-01
Equations built on fractional derivatives prove to be a powerful tool in the description of complex systems when the effects of singularity, fractal supports, and long-range dependence play a role. In this Letter, we advocate an application of the fractional derivative formalism to a fairly general...... class of critical phenomena when the organization of the system near the phase transition point is influenced by a competing nonlocal ordering. Fractional modifications of the free energy functional at criticality and of the widely known Ginzburg-Landau equation central to the classical Landau theory...... of second-type phase transitions are discussed in some detail. An implication of the fractional Ginzburg-Landau equation is a renormalization of the transition temperature owing to the nonlocality present. (c) 2005 Elsevier B.V. All rights reserved....
Cognitive Load in Algebra: Element Interactivity in Solving Equations
Ngu, Bing Hiong; Chung, Siu Fung; Yeung, Alexander Seeshing
2015-01-01
Central to equation solving is the maintenance of equivalence on both sides of the equation. However, when the process involves an interaction of multiple elements, solving an equation can impose a high cognitive load. The balance method requires operations on both sides of the equation, whereas the inverse method involves operations on one side…
Solving Kepler's equation using implicit functions
Mortari, Daniele; Elipe, Antonio
2014-01-01
A new approach to solve Kepler's equation based on the use of implicit functions is proposed here. First, new upper and lower bounds are derived for two ranges of mean anomaly. These upper and lower bounds initialize a two-step procedure involving the solution of two implicit functions. These two implicit functions, which are non-rational (polynomial) Bézier functions, can be linear or quadratic, depending on the derivatives of the initial bound values. These are new initial bounds that have been compared and proven more accurate than Serafin's bounds. The procedure reaches machine error accuracy with no more that one quadratic and one linear iterations, experienced in the "tough range", where the eccentricity is close to one and the mean anomaly to zero. The proposed method is particularly suitable for space-based applications with limited computational capability.
Application of the trial equation method for solving some nonlinear ...
Indian Academy of Sciences (India)
Their key idea is to expand solutions of given differential equations as functions of solutions of solvable differential equations, in particular, polynomial and rational func- tions. This idea is so good that many types of nonlinear differential equations can be solved by it. Also, Liu proposed the trial equation method and applied ...
Sinc-collocation method for solving the Blasius equation
International Nuclear Information System (INIS)
Parand, K.; Dehghan, Mehdi; Pirkhedri, A.
2009-01-01
Sinc-collocation method is applied for solving Blasius equation which comes from boundary layer equations. It is well known that sinc procedure converges to the solution at an exponential rate. Comparison with Howarth and Asaithambi's numerical solutions reveals that the proposed method is of high accuracy and reduces the solution of Blasius' equation to the solution of a system of algebraic equations.
Soliton equations solved by the boundary CFT
Saito, Satoru; Sato, Ryuichi
2003-01-01
Soliton equations are derived which characterize the boundary CFT a la Callan et al. Soliton fields of classical soliton equations are shown to appear as a neutral bound state of a pair of soliton fields of BCFT. One soliton amplitude under the influence of the boundary is calculated explicitly and is shown that it is frozen at the Dirichlet limit.
Variational iteration method for solving coupled-KdV equations
International Nuclear Information System (INIS)
Assas, Laila M.B.
2008-01-01
In this paper, the He's variational iteration method is applied to solve the non-linear coupled-KdV equations. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. This technique provides a sequence of functions which converge to the exact solution of the coupled-KdV equations. This procedure is a powerful tool for solving coupled-KdV equations
Solving the Noncommutative Batalin-Vilkovisky Equation
Barannikov, Serguei
2013-06-01
Given an odd symmetry acting on an associative algebra, I show that the summation over arbitrary ribbon graphs gives the construction of the solutions to the noncommutative Batalin-Vilkovisky equation, introduced in (Barannikov in IMRN, rnm075, 2007), and to the equivariant version of this equation. This generalizes the known construction of A ∞-algebra via summation over ribbon trees. I give also the generalizations to other types of algebras and graph complexes, including the stable ribbon graph complex. These solutions to the noncommutative Batalin-Vilkovisky equation and to its equivariant counterpart, provide naturally the supersymmetric matrix action functionals, which are the gl( N)-equivariantly closed differential forms on the matrix spaces, as in (Barannikov in Comptes Rendus Mathematique vol 348, pp. 359-362.
An approach to numerically solving the Poisson equation
Feng, Zhichen; Sheng, Zheng-Mao
2015-06-01
We introduce an approach for numerically solving the Poisson equation by using a physical model, which is a way to solve a partial differential equation without the finite difference method. This method is especially useful for obtaining the solutions in very many free-charge neutral systems with open boundary conditions. It can be used for arbitrary geometry and mesh style and is more efficient comparing with the widely-used iterative algorithm with multigrid methods. It is especially suitable for parallel computing. This method can also be applied to numerically solving other partial differential equations whose Green functions exist in analytic expression.
An efficient numerical method for solving nonlinear foam drainage equation
Parand, Kourosh; Delkhosh, Mehdi
2018-02-01
In this paper, the nonlinear foam drainage equation, which is a famous nonlinear partial differential equation, is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized fractional order of the Chebyshev functions (B-GFCF) collocation method. First, using the quasilinearization method, the equation is converted into a sequence of linear partial differential equations (LPD), and then these LPDs are solved using the B-GFCF collocation method. A very good approximation of solutions is obtained, and comparisons show that the obtained results are more accurate than the results of other researchers.
Conversational eigenanalysis program for solving differential equations
International Nuclear Information System (INIS)
Killough, G.G.; Eckerman, K.F.
1984-01-01
Dynamic models that arise in health physics applications are often expressed in terms of systems of ordinary differential equations. In many cases, such as box models that describe material exchange among reservoirs, the differential equations are linear with constant coefficients, and the analysis can be reduced to the examination of solutions of initial-value problems for such systems. This paper describes a conversational code, DIFSOL, that permits the user to specify the coefficient matrix and an initial vector of the system; DIFSOL prints out closed-form solutions [i.e., expressed as linear combinations of terms of the form e/sup -at/, e/sup -at/cos bt, and e/sup -at/sin bt] and tables of the solution, its derivative, and its integral for any specified linear combination of state variables. The program logic permits menu-driven control. We have operated a FORTRAN IV version of the code on a DEC PDP-10 for several years. A translation into BASIC has proved practical on Radio Shack TRS-80 Model I and III personal computers for smaller systems of differential equations (< 12 state variables). The paper includes illustrations of the use of DIFSOL in studying metabolic models
Final report [on solving the multigroup diffusion equations
International Nuclear Information System (INIS)
Birkhoff, G.
1975-01-01
Progress achieved in the development of variational methods for solving the multigroup neutron diffusion equations is described. An appraisal is made of the extent to which improved variational methods could advantageously replace difference methods currently used
Solving Variable Coefficient Fourth-Order Parabolic Equation by ...
African Journals Online (AJOL)
In this paper, a Modified initial guess Variational Iteration Method (MigVIM) is used to solve a non-homogeneous variable coefficient fourth order parabolic partial differential equations. The new method shows rapid convergence to the exact solution.
An Efficient Algorithm for Solving the Telegraph Equation ...
African Journals Online (AJOL)
Abstract. In this work, we propose a numerical scheme to solve telegraph equations using modified variational iteration method. The numerical results are compared with analytical solutions to confirm the efficiency of the method.
A new Newton-like method for solving nonlinear equations.
Saheya, B; Chen, Guo-Qing; Sui, Yun-Kang; Wu, Cai-Ying
2016-01-01
This paper presents an iterative scheme for solving nonline ar equations. We establish a new rational approximation model with linear numerator and denominator which has generalizes the local linear model. We then employ the new approximation for nonlinear equations and propose an improved Newton's method to solve it. The new method revises the Jacobian matrix by a rank one matrix each iteration and obtains the quadratic convergence property. The numerical performance and comparison show that the proposed method is efficient.
students' preference of method of solving simultaneous equations
African Journals Online (AJOL)
Ugboduma,Samuel.O.
More so, they will be reposition in order to solve other mathematical problems involving applications of simultaneous equations methods of solutions. Research questions. The following research questions were asked: i what proportion of SSS2 students showed preference for one type of method of solving simultaneous.
Adams Predictor-Corrector Systems for Solving Fuzzy Differential Equations
Directory of Open Access Journals (Sweden)
Dequan Shang
2013-01-01
Full Text Available A predictor-corrector algorithm and an improved predictor-corrector (IPC algorithm based on Adams method are proposed to solve first-order differential equations with fuzzy initial condition. These algorithms are generated by updating the Adams predictor-corrector method and their convergence is also analyzed. Finally, the proposed methods are illustrated by solving an example.
DEVELOPMENT OF TEACHING MATERIALS ALGEBRAIC EQUATION TO IMPROVE PROBLEM SOLVING
Directory of Open Access Journals (Sweden)
sri adi widodo
2017-01-01
Full Text Available Problem-solving skills are the basic capabilities of a person in solving a problem and that involve critical thinking, logical, and systematic. To solve a problem one-way necessary measures to solve the problem. Polya is one way to solve a mathematical problem. by developing teaching materials designed using the steps in solving problems Polya expected students could improve its ability to solve problems. In this first year, the goal of this study is to investigate the process of learning the hypothetical development of teaching materials. This study is a research & development. Procedure development research refers to research the development of Thiagarajan, Semmel & Semmel ie 4-D. Model development in the first year is define, design, and development. The collection of data for the assessment of teaching materials algebra equations conducted by the expert by filling the validation sheet. Having examined the materials of algebraic equations in the subject of numerical methods, reviewing the curriculum that is aligned with KKNI, and formulates learning outcomes that formed the conceptual teaching material on the material algebraic equations. From the results of expert assessment team found that the average ratings of teaching materials in general algebraic equation of 4.38 with a very good category. The limited test needs to be done to see effectiveness teaching materials on problem-solving skills in students who are taking courses numerical methods
Solving Abel integral equations of first kind via fractional calculus
Directory of Open Access Journals (Sweden)
Salman Jahanshahi
2015-04-01
Full Text Available We give a new method for numerically solving Abel integral equations of first kind. An estimation for the error is obtained. The method is based on approximations of fractional integrals and Caputo derivatives. Using trapezoidal rule and Computer Algebra System Maple, the exact and approximation values of three Abel integral equations are found, illustrating the effectiveness of the proposed approach.
The perturbation method to solve subdiffusion-reaction equations
Lewandowska, Katarzyna D.; Kosztołowicz, Tadeusz; Piwnik, Mateusz
2012-01-01
We use the perturbation method to approximately solve subdiffusion-reaction equations. Within this method we obtain the solutions of the zeroth and the first order. The comparison our analytical solutions with the numerical results shown that the perturbation method can be useful to find approximate solutions of nonlinear subdiffusion--reaction equations.
Students' errors in solving linear equation word problems: Case ...
African Journals Online (AJOL)
The study examined errors students make in solving linear equation word problems with a view to expose the nature of these errors and to make suggestions for classroom teaching. A diagnostic test comprising 10 linear equation word problems, was administered to a sample (n=130) of senior high school first year Home ...
Solving wave equation using finite differences and Taylor series
Nečasová, Gabriela; Kocina, Filip; Veigend, Petr; Chaloupka, Jan; Šátek, Václav; Kunovský, Jiří
2017-07-01
The paper deals with the numerical solution of partial differential equations (PDEs), especially wave equation. Two methods are used to obtain numerical solution of the wave equation. The Finite Difference Method (FDM) is used for transformation of wave equation to the system of ordinary differential equations (ODEs), different types of difference formulas are used. The influence of arithmetic to higher order difference formulas is also presented. The Modern Taylor Series Method (MTSM) allows to solve ODEs numerically with extremely high precision. An important feature of this method is an automatic integration order setting, i.e. using as many Taylor series terms as the defined accuracy requires.
Rational Chebyshev pseudospectral approach for solving Thomas-Fermi equation
Energy Technology Data Exchange (ETDEWEB)
Parand, K. [Department of Computer Sciences, Shahid Beheshti University, Tehran (Iran, Islamic Republic of)], E-mail: k_parand@sbu.ac.ir; Shahini, M. [Department of Computer Sciences, Shahid Beheshti University, Tehran (Iran, Islamic Republic of)], E-mail: m.shahini@mail.sbu.ac.ir
2009-01-05
In this Letter we propose a pseudospectral method for solving Thomas-Fermi equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on rational Chebyshev pseudospectral method. This method reduces the solution of this problem to the solution of a system of algebraic equations. Comparison with some numerical solutions shows that the present solution is highly accurate.
Nodal spectrum method for solving neutron diffusion equation
International Nuclear Information System (INIS)
Sanchez, D.; Garcia, C. R.; Barros, R. C. de; Milian, D.E.
1999-01-01
Presented here is a new numerical nodal method for solving static multidimensional neutron diffusion equation in rectangular geometry. Our method is based on a spectral analysis of the nodal diffusion equations. These equations are obtained by integrating the diffusion equation in X, Y directions and then considering flat approximations for the current. These flat approximations are the only approximations that are considered in this method, as a result the numerical solutions are completely free from truncation errors. We show numerical results to illustrate the methods accuracy for coarse mesh calculations
Wavelet Methods for Solving Fractional Order Differential Equations
Directory of Open Access Journals (Sweden)
A. K. Gupta
2014-01-01
Full Text Available Fractional calculus is a field of applied mathematics which deals with derivatives and integrals of arbitrary orders. The fractional calculus has gained considerable importance during the past decades mainly due to its application in diverse fields of science and engineering such as viscoelasticity, diffusion of biological population, signal processing, electromagnetism, fluid mechanics, electrochemistry, and many more. In this paper, we review different wavelet methods for solving both linear and nonlinear fractional differential equations. Our goal is to analyze the selected wavelet methods and assess their accuracy and efficiency with regard to solving fractional differential equations. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on various wavelets in order to solve differential equations of arbitrary order.
Solving Systems of Equations with Techniques from Artificial Intelligence
Directory of Open Access Journals (Sweden)
Irina Maria Terfaloaga
2015-07-01
Full Text Available A frequent problem in numerical analysis is solving the systems of equations. That problem has generated in time a great interest among mathematicians and computer scientists, as evidenced by the large number of numerical methods developed. Besides the classical numerical methods, in the last years were proposed methods inspired by techniques from artificial intelligence. Hybrid methods have been also proposed along the time [15, 19]. The goal of this study is to make a survey of methods inspired from artificial intelligence for solving systems of equations
Multiparameter extrapolation and deflation methods for solving equation systems
Directory of Open Access Journals (Sweden)
A. J. Hughes Hallett
1984-01-01
Full Text Available Most models in economics and the applied sciences are solved by first order iterative techniques, usually those based on the Gauss-Seidel algorithm. This paper examines the convergence of multiparameter extrapolations (accelerations of first order iterations, as an improved approximation to the Newton method for solving arbitrary nonlinear equation systems. It generalises my earlier results on single parameter extrapolations. Richardson's generalised method and the deflation method for detecting successive solutions in nonlinear equation systems are also presented as multiparameter extrapolations of first order iterations. New convergence results are obtained for those methods.
Approximate Method for Solving the Linear Fuzzy Delay Differential Equations
Directory of Open Access Journals (Sweden)
S. Narayanamoorthy
2015-01-01
Full Text Available We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.
Numerical Solution of Turbulence Problems by Solving Burgers’ Equation
Directory of Open Access Journals (Sweden)
Alicia Cordero
2015-05-01
Full Text Available In this work we generate the numerical solutions of Burgers’ equation by applying the Crank-Nicholson method and different schemes for solving nonlinear systems, instead of using Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. The method is analyzed on two test problems in order to check its efficiency on different kinds of initial conditions. Numerical solutions as well as exact solutions for different values of viscosity are calculated, concluding that the numerical results are very close to the exact solution.
Modified Chebyshev Collocation Method for Solving Differential Equations
Directory of Open Access Journals (Sweden)
M Ziaul Arif
2015-05-01
Full Text Available This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial collocation method is applied to both Ordinary Differential Equations (ODEs and Partial Differential Equations (PDEs cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.
A neuro approach to solve fuzzy Riccati differential equations
Energy Technology Data Exchange (ETDEWEB)
Shahrir, Mohammad Shazri, E-mail: mshazri@gmail.com [InstitutSainsMatematik, Universiti Malaya 50603 Kuala Lumpur, Wilayah Persekutuan Kuala Lumpur (Malaysia); Telekom Malaysia, R& D TM Innovation Centre, LingkaranTeknokrat Timur, 63000 Cyberjaya, Selangor (Malaysia); Kumaresan, N., E-mail: drnk2008@gmail.com; Kamali, M. Z. M.; Ratnavelu, Kurunathan [InstitutSainsMatematik, Universiti Malaya 50603 Kuala Lumpur, Wilayah Persekutuan Kuala Lumpur (Malaysia)
2015-10-22
There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.
He's iteration formulation for solving nonlinear algebraic equations
International Nuclear Information System (INIS)
Qian, W-X; Ye, Y-H; Chen, J; Mo, L-F
2008-01-01
Newton iteration method is sensitive to initial guess and its slope. To overcome the shortcoming, He's iteration method is used to solve nonlinear algebraic equations where Newton iteration method becomes invalid. Some examples are given, showing that the method is effective
A Simple Derivation of Kepler's Laws without Solving Differential Equations
Provost, J.-P.; Bracco, C.
2009-01-01
Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non-trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple…
Students' Preference of Method of Solving Simultaneous Equations ...
African Journals Online (AJOL)
The study focused on students' preference of two methods of solving simultaneous equations. The researcher adopted three research questions and three hypotheses. The research questions were answered using the descriptive statistics of percentages, while, the hypotheses were tested using the chi-square. The study ...
Numerical Aspects of Solving Differential Equations: Laboratory Approach for Students.
Witt, Ana
1997-01-01
Describes three labs designed to help students in a first course on ordinary differential equations with three of the most common numerical difficulties they might encounter when solving initial value problems with a numerical software package. The goal of these labs is to help students advance to independent work on common numerical anomalies.…
A novel approach for solving fractional Fisher equation using ...
Indian Academy of Sciences (India)
same equation occurs in logistic population growth models, flame propagation, neuro- physiology, autocatalytic chemical reactions and branching Brownian motion processes. The differential transform method was first introduced by Zhou [8] who solved linear and nonlinear initial value problems in electric circuit analysis.
Insights into the School Mathematics Tradition from Solving Linear Equations
Buchbinder, Orly; Chazan, Daniel; Fleming, Elizabeth
2015-01-01
In this article, we explore how the solving of linear equations is represented in English-language algebra text books from the early nineteenth century when schooling was becoming institutionalized, and then survey contemporary teachers. In the text books, we identify the increasing presence of a prescribed order of steps (a canonical method) for…
Ten-Year-Old Students Solving Linear Equations
Brizuela, Barbara; Schliemann, Analucia
2004-01-01
In this article, the authors seek to re-conceptualize the perspective regarding students' difficulties with algebra. While acknowledging that students "do" have difficulties when learning algebra, they also argue that the generally espoused criteria for algebra as the ability to work with the syntactical rules for solving equations is…
Experimental quantum computing to solve systems of linear equations.
Cai, X-D; Weedbrook, C; Su, Z-E; Chen, M-C; Gu, Mile; Zhu, M-J; Li, Li; Liu, Nai-Le; Lu, Chao-Yang; Pan, Jian-Wei
2013-06-07
Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time proportional to the number of variables N. A recently proposed quantum algorithm shows that quantum computers could solve linear systems in a time scale of order log(N), giving an exponential speedup over classical computers. Here we realize the simplest instance of this algorithm, solving 2×2 linear equations for various input vectors on a quantum computer. We use four quantum bits and four controlled logic gates to implement every subroutine required, demonstrating the working principle of this algorithm.
Measures of Potential Flexibility and Practical Flexibility in Equation Solving
Directory of Open Access Journals (Sweden)
Le Xu
2017-08-01
Full Text Available Researchers interested in mathematical proficiency have recently begun to explore the development of strategic flexibility, where flexibility is defined as knowledge of multiple strategies for solving a problem and the ability to implement an innovative strategy for a given problem solving circumstance. However, anecdotal findings from this literature indicate that students do not consistently use an innovative strategy for solving a given problem, even when these same students demonstrate knowledge of innovative strategies. This distinction, sometimes framed in the psychological literature as competence vs. performance—has not been previously studied for flexibility. In order to explore the competence/performance distinction in flexibility, this study developed and validated measures for potential flexibility (e.g., competence, or knowledge of multiple strategies and practical flexibility (e.g., performance, use of innovative strategies for solving equations. The measures were administrated to a sample of 158 Chinese middle school students through a Tri-Phase Flexibility Assessment, in which the students were asked to solve each equation, generate additional strategies, and evaluate own multiple strategies. Confirmatory factor analysis supported a two-factor model of potential and practical flexibility. Satisfactory internal consistency was found for the measures. Additional validity evidence included the significant association with flexibility measured with the previous method. Potential flexibility and practical flexibility were found to be distinct but related. The theoretical and practical implications of the concepts and their measures of potential flexibility and practical flexibility are discussed.
A method of solving simple harmonic oscillator Schroedinger equation
Maury, Juan Carlos F.
1995-01-01
A usual step in solving totally Schrodinger equation is to try first the case when dimensionless position independent variable w is large. In this case the Harmonic Oscillator equation takes the form (d(exp 2)/dw(exp 2) - w(exp 2))F = 0, and following W.K.B. method, it gives the intermediate corresponding solution F = exp(-w(exp 2)/2), which actually satisfies exactly another equation, (d(exp 2)/dw(exp 2) + 1 - w(exp 2))F = 0. We apply a different method, useful in anharmonic oscillator equations, similar to that of Rampal and Datta, and although it is slightly more complicated however it is also more general and systematic.
An efficient method for solving the steady Euler equations
Liou, M.-S.
1986-01-01
An efficient numerical procedure for solving a set of nonlinear partial differential equations, the steady Euler equations, using Newton's linearization procedure is presented. A theorem indicating quadratic convergence for the case of differential equations is demonstrated. A condition for the domain of quadratic convergence Omega(2) is obtained which indicates that whether an approximation lies in Omega(2) depends on the rate of change and the smoothness of the flow vectors, and hence is problem-dependent. The choice of spatial differencing, of particular importance for the present method, is discussed. The treatment of boundary conditions is addressed, and the system of equations resulting from the foregoing analysis is summarized and solution strategies are discussed. The convergence of calculated solutions is demonstrated by comparing them with exact solutions to one and two-dimensional problems.
Artificial neural networks for solving ordinary and partial differential equations.
Lagaris, I E; Likas, A; Fotiadis, D I
1998-01-01
We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE's), to systems of coupled ODE's and also to partial differential equations (PDE's). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galekrkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.
Solving the stationary Liouville equation via a boundary element method
Chappell, David J.; Tanner, Gregor
2013-02-01
Energy distributions of linear wave fields are, in the high frequency limit, often approximated in terms of flow or transport equations in phase space. Common techniques for solving the flow equations in both time dependent and stationary problems are ray tracing and level set methods. In the context of predicting the vibro-acoustic response of complex engineering structures, related methods such as Statistical Energy Analysis or variants thereof have found widespread applications. We present a new method for solving the transport equations for complex multi-component structures based on a boundary element formulation of the stationary Liouville equation. The method is an improved version of the Dynamical Energy Analysis technique introduced recently by the authors. It interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. We demonstrate that the method can be used to efficiently deal with complex large scale problems giving good approximations of the energy distribution when compared to exact solutions of the underlying wave equation.
Various Newton-type iterative methods for solving nonlinear equations
Directory of Open Access Journals (Sweden)
Manoj Kumar
2013-10-01
Full Text Available The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.
Solving Nonlinear Partial Differential Equations with Maple and Mathematica
Shingareva, Inna K
2011-01-01
The emphasis of the book is given in how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple an
An Unconditionally Stable Method for Solving the Acoustic Wave Equation
Directory of Open Access Journals (Sweden)
Zhi-Kai Fu
2015-01-01
Full Text Available An unconditionally stable method for solving the time-domain acoustic wave equation using Associated Hermit orthogonal functions is proposed. The second-order time derivatives in acoustic wave equation are expanded by these orthogonal basis functions. By applying Galerkin temporal testing procedure, the time variable can be eliminated from the calculations. The restriction of Courant-Friedrichs-Levy (CFL condition in selecting time step for analyzing thin layer can be avoided. Numerical results show the accuracy and the efficiency of the proposed method.
A Meshfree Quasi-Interpolation Method for Solving Burgers’ Equation
Directory of Open Access Journals (Sweden)
Mingzhu Li
2014-01-01
Full Text Available The main aim of this work is to consider a meshfree algorithm for solving Burgers’ equation with the quartic B-spline quasi-interpolation. Quasi-interpolation is very useful in the study of approximation theory and its applications, since it can yield solutions directly without the need to solve any linear system of equations and overcome the ill-conditioning problem resulting from using the B-spline as a global interpolant. The numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. Compared to other numerical methods, the main advantages of our scheme are higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.
New numerical method for solving the solute transport equation
International Nuclear Information System (INIS)
Ross, B.; Koplik, C.M.
1978-01-01
The solute transport equation can be solved numerically by approximating the water flow field by a network of stream tubes and using a Green's function solution within each stream tube. Compared to previous methods, this approach permits greater computational efficiency and easier representation of small discontinuities, and the results are easier to interpret physically. The method has been used to study hypothetical sites for disposal of high-level radioactive waste
Variational Iteration Method for Solving a Fuzzy Generalized Pantograph Equation
Directory of Open Access Journals (Sweden)
A. Amiri
2014-05-01
Full Text Available A numerical method for solving the fuzzy generalized pantograph equation under fuzzy initial value conditions is presented. This technique provides a sequence of functions which converges to the exact solution to the problem and is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. To display the validity and applicability of the numerical method two illustrative examples are presented.
The Tdgl Equation for Car-Following Model with Consideration of the Traffic Interruption Probability
Ge, Hong-Xia; Zhang, Yi-Qiang; Kuang, Hua; Lo, Siu-Ming
2012-07-01
A car-following model which involves the effects of traffic interruption probability is further investigated. The stability condition of the model is obtained through the linear stability analysis. The reductive perturbation method is taken to derive the time-dependent Ginzburg-Landau (TDGL) equation to describe the traffic flow near the critical point. Moreover, the coexisting curve and the spinodal line are obtained by the first and second derivatives of the thermodynamic potential, respectively. The analytical results show that considering the interruption effects could further stabilize traffic flow.
Polynomial mixture method of solving ordinary differential equations
Shahrir, Mohammad Shazri; Nallasamy, Kumaresan; Ratnavelu, Kuru; Kamali, M. Z. M.
2017-11-01
In this paper, a numerical solution of fuzzy quadratic Riccati differential equation is estimated using a proposed new approach that provides mixture of polynomials where iteratively the right mixture will be generated. This mixture provide a generalized formalism of traditional Neural Networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). This can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that Polynomial Mixture Method (PMM) shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over Mabood et al, RK-4, Multi-Agent NN and Neuro Method (NM).
Encouraging Students to Think Strategically when Learning to Solve Linear Equations
Robson, Daphne; Abell, Walt; Boustead, Therese
2012-01-01
Students who are preparing to study science and engineering need to understand equation solving but adult students returning to study can find this difficult. In this paper, the design of an online resource, Equations2go, for helping students learn to solve linear equations is investigated. Students learning to solve equations need to consider…
Mathematics Literacy of Secondary Students in Solving Simultanenous Linear Equations
Sitompul, R. S. I.; Budayasa, I. K.; Masriyah
2018-01-01
This study examines the profile of secondary students’ mathematical literacy in solving simultanenous linear equations problems in terms of cognitive style of visualizer and verbalizer. This research is a descriptive research with qualitative approach. The subjects in this research consist of one student with cognitive style of visualizer and one student with cognitive style of verbalizer. The main instrument in this research is the researcher herself and supporting instruments are cognitive style tests, mathematics skills tests, problem-solving tests and interview guidelines. Research was begun by determining the cognitive style test and mathematics skill test. The subjects chosen were given problem-solving test about simultaneous linear equations and continued with interview. To ensure the validity of the data, the researcher conducted data triangulation; the steps of data reduction, data presentation, data interpretation, and conclusion drawing. The results show that there is a similarity of visualizer and verbalizer-cognitive style in identifying and understanding the mathematical structure in the process of formulating. There are differences in how to represent problems in the process of implementing, there are differences in designing strategies and in the process of interpreting, and there are differences in explaining the logical reasons.
Solving Systems of Linear Equations with a Superconducting Quantum Processor.
Zheng, Yarui; Song, Chao; Chen, Ming-Cheng; Xia, Benxiang; Liu, Wuxin; Guo, Qiujiang; Zhang, Libo; Xu, Da; Deng, Hui; Huang, Keqiang; Wu, Yulin; Yan, Zhiguang; Zheng, Dongning; Lu, Li; Pan, Jian-Wei; Wang, H; Lu, Chao-Yang; Zhu, Xiaobo
2017-05-26
Superconducting quantum circuits are a promising candidate for building scalable quantum computers. Here, we use a four-qubit superconducting quantum processor to solve a two-dimensional system of linear equations based on a quantum algorithm proposed by Harrow, Hassidim, and Lloyd [Phys. Rev. Lett. 103, 150502 (2009)PRLTAO0031-900710.1103/PhysRevLett.103.150502], which promises an exponential speedup over classical algorithms under certain circumstances. We benchmark the solver with quantum inputs and outputs, and characterize it by nontrace-preserving quantum process tomography, which yields a process fidelity of 0.837±0.006. Our results highlight the potential of superconducting quantum circuits for applications in solving large-scale linear systems, a ubiquitous task in science and engineering.
Solving the generalized Langevin equation with the algebraically correlated noise
International Nuclear Information System (INIS)
Srokowski, T.; Ploszajczak, M.
1997-01-01
The Langevin equation with the memory kernel is solved. The stochastic force possesses algebraic correlations, proportional to 1/t. The velocity autocorrelation function and related quantities characterizing transport properties are calculated at the assumption that the system is in the thermal equilibrium. Stochastic trajectories are simulated numerically, using the kangaroo process as a noise generator. Results of this simulation resemble Levy walks with divergent moments of the velocity distribution. The motion of a Brownian particle is considered both without any external potential and in the harmonic oscillator field, in particular the escape from a potential well. The results are compared with memory-free calculations for the Brownian particle. (author)
A rational function based scheme for solving advection equation
Energy Technology Data Exchange (ETDEWEB)
Xiao, Feng [Gunma Univ., Kiryu (Japan). Faculty of Engineering; Yabe, Takashi
1995-07-01
A numerical scheme for solving advection equations is presented. The scheme is derived from a rational interpolation function. Some properties of the scheme with respect to convex-concave preserving and monotone preserving are discussed. We find that the scheme is attractive in surpressinging overshoots and undershoots even in the vicinities of discontinuity. The scheme can also be easily swicthed as the CIP (Cubic interpolated Pseudo-Particle) method to get a third-order accuracy in smooth region. Numbers of numerical tests are carried out to show the non-oscillatory and less diffusive nature of the scheme. (author).
A Sumudu based algorithm for solving differential equations
Directory of Open Access Journals (Sweden)
Jun Zhang
2007-11-01
Full Text Available An algorithm based on Sumudu transform is developed. The algorithm can be implemented in computer algebra systems like Maple. It can be used to solve differential equations of the following form automatically without human interaction \\begin{displaymath} \\sum_{i=0}^{m} p_i(xy^{(i}(x = \\sum_{j=0}^{k}q_j(xh_j(x \\end{displaymath} where pi(x(i=0, 1, 2, ..., m and qj(x(j=0, 1, 2, ..., k are polynomials. hj(x are non-rational functions, but their Sumudu transforms are rational. m, k are nonnegative integers.
A simple derivation of Kepler's laws without solving differential equations
International Nuclear Information System (INIS)
Provost, J-P; Bracco, C
2009-01-01
Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non-trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple reconsideration of Newton's figure naturally leads to an explicit expression of the velocity and to the equation of the trajectory. This derivation, which can be fully apprehended by undergraduates or by secondary school teachers (who might use it with their pupils), can be considered as a first application of mechanical concepts to a physical problem of great historical and pedagogical interest
Solving the quantum brachistochrone equation through differential geometry
You, Chenglong; Dowling, Jonathan; Wang, Xiaoting
The ability of generating a particular quantum state, or model a physical quantum device by exploring quantum state transfer, is important in many applications. Due to the environmental noise, a quantum device suffers from decoherence causing information loss. Hence, completing the state-generation task in a time-optimal way can be considered as a straightforward method to reduce decoherence. For a quantum system whose Hamiltonian has a fixed type and a finite energy bandwidth, it has been found that the time-optimal quantum evolution can be characterized by the quantum brachistochrone equation. In addition, the brachistochrone curve is found to have a geometric interpretation: it is the limit of a one-parameter family of geodesics on a sub-Riemannian model. Such geodesic-brachistochrone connection provides an efficient numerical method to solve the quantum brachistochrone equation. In this work, we will demonstrate this numerical method by studying the time-optimal state-generating problem on a given quantum spin system.We also find that the Pareto weighted-sum optimization turns out to be a simple but efficient method in solving the quantum time-optimal problems.
[Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (2)].
Murase, Kenya
2015-01-01
In this issue, symbolic methods for solving differential equations were firstly introduced. Of the symbolic methods, Laplace transform method was also introduced together with some examples, in which this method was applied to solving the differential equations derived from a two-compartment kinetic model and an equivalent circuit model for membrane potential. Second, series expansion methods for solving differential equations were introduced together with some examples, in which these methods were used to solve Bessel's and Legendre's differential equations. In the next issue, simultaneous differential equations and various methods for solving these differential equations will be introduced together with some examples in medical physics.
Modeling Blazar Spectra by Solving an Electron Transport Equation
Lewis, Tiffany; Finke, Justin; Becker, Peter A.
2018-01-01
Blazars are luminous active galaxies across the entire electromagnetic spectrum, but the spectral formation mechanisms, especially the particle acceleration, in these sources are not well understood. We develop a new theoretical model for simulating blazar spectra using a self-consistent electron number distribution. Specifically, we solve the particle transport equation considering shock acceleration, adiabatic expansion, stochastic acceleration due to MHD waves, Bohm diffusive particle escape, synchrotron radiation, and Compton radiation, where we implement the full Compton cross-section for seed photons from the accretion disk, the dust torus, and 26 individual broad lines. We used a modified Runge-Kutta method to solve the 2nd order equation, including development of a new mathematical method for normalizing stiff steady-state ordinary differential equations. We show that our self-consistent, transport-based blazar model can qualitatively fit the IR through Fermi g-ray data for 3C 279, with a single-zone, leptonic configuration. We use the solution for the electron distribution to calculate multi-wavelength SED spectra for 3C 279. We calculate the particle and magnetic field energy densities, which suggest that the emitting region is not always in equipartition (a common assumption), but sometimes matter dominated. The stratified broad line region (based on ratios in quasar reverberation mapping, and thus adding no free parameters) improves our estimate of the location of the emitting region, increasing it by ~5x. Our model provides a novel view into the physics at play in blazar jets, especially the relative strength of the shock and stochastic acceleration, where our model is well suited to distinguish between these processes, and we find that the latter tends to dominate.
An inherently parallel method for solving discretized diffusion equations
International Nuclear Information System (INIS)
Eccleston, B.R.; Palmer, T.S.
1999-01-01
A Monte Carlo approach to solving linear systems of equations is being investigated in the context of the solution of discretized diffusion equations. While the technique was originally devised decades ago, changes in computer architectures (namely, massively parallel machines) have driven the authors to revisit this technique. There are a number of potential advantages to this approach: (1) Analog Monte Carlo techniques are inherently parallel; this is not necessarily true to today's more advanced linear equation solvers (multigrid, conjugate gradient, etc.); (2) Some forms of this technique are adaptive in that they allow the user to specify locations in the problem where resolution is of particular importance and to concentrate the work at those locations; and (3) These techniques permit the solution of very large systems of equations in that matrix elements need not be stored. The user could trade calculational speed for storage if elements of the matrix are calculated on the fly. The goal of this study is to compare the parallel performance of Monte Carlo linear solvers to that of a more traditional parallelized linear solver. The authors observe the linear speedup that they expect from the Monte Carlo algorithm, given that there is no domain decomposition to cause significant communication overhead. Overall, PETSc outperforms the Monte Carlo solver for the test problem. The PETSc parallel performance improves with larger numbers of unknowns for a given number of processors. Parallel performance of the Monte Carlo technique is independent of the size of the matrix and the number of processes. They are investigating modifications to the scheme to accommodate matrix problems with positive off-diagonal elements. They are also currently coding an on-the-fly version of the algorithm to investigate the solution of very large linear systems
Simonucci, S.; Strinati, G. C.
2014-02-01
We derive a nonlinear differential equation for the gap parameter of a superfluid Fermi system by performing a suitable coarse graining of the Bogoliubov-de Gennes (BdG) equations throughout the BCS-BEC crossover, with the aim of replacing the time-consuming solution of the original BdG equations by the simpler solution of this novel equation. We perform a favorable numerical test on the validity of this new equation over most of the temperature-coupling phase diagram, by an explicit comparison with the full solution of the original BdG equations for an isolated vortex. We also show that the new equation reduces both to the Ginzburg-Landau equation for Cooper pairs in weak coupling close to the critical temperature and to the Gross-Pitaevskii equation for composite bosons in strong coupling at low temperature.
Numerical method for solving integral equations of neutron transport. II
International Nuclear Information System (INIS)
Loyalka, S.K.; Tsai, R.W.
1975-01-01
In a recent paper it was pointed out that the weakly singular integral equations of neutron transport can be quite conveniently solved by a method based on subtraction of singularity. This previous paper was devoted entirely to the consideration of simple one-dimensional isotropic-scattering and one-group problems. The present paper constitutes interesting extensions of the previous work in that in addition to a typical two-group anisotropic-scattering albedo problem in the slab geometry, the method is also applied to an isotropic-scattering problem in the x-y geometry. These results are compared with discrete S/sub N/ (ANISN or TWOTRAN-II) results, and for the problems considered here, the proposed method is found to be quite effective. Thus, the method appears to hold considerable potential for future applications. (auth)
An iterative method for solving neutron transport equation
International Nuclear Information System (INIS)
Simovic, R.
1988-01-01
Assuming a plane geometry and isotropic form of the neutron scattering function a new iterative method for solving the one-velocity transport equation is developed. The basic point of this method is the definition of the neutron fluxes Φ n± (x, μ, μ 0 ) representing the space dependent angular distributions of neutrons scattered n-times in directions μ 0. This makes possible to construct a new system for successive calculation of Φ n± (x, μ, μ 0 ) starting with the flux of un-collided neutrons. This treatment was shown to be more efficient than the ordinary one. As examples, the infinite medium Green functions and reflection coefficients of half space were calculated and analyzed. (author)
Numerical method for solving stochastic differential equations with dichotomous noise.
Kim, Changho; Lee, Eok Kyun; Talkner, Peter
2006-02-01
We propose a numerical method for solving stochastic differential equations with dichotomous Markov noise. The numerical scheme is formulated such that (i) the stochastic formula used follows the Stratonovich-Taylor form over the entire range of noise correlation times, including the Gaussian white noise limit; and (ii) the method is readily applicable to dynamical systems driven by arbitrary types of noise, provided there exists a way to describe the random increment of the stochastic process expressed in the Stratonovich-Taylor form. We further propose a simplified Taylor scheme that significantly reduces the computation time, while still satisfying the moment properties up to the required order. The accuracies and efficiencies of the proposed algorithms are validated by applying the schemes to two prototypical model systems that possess analytical solutions.
Solving the power flow equations: a monotone operator approach
Energy Technology Data Exchange (ETDEWEB)
Dvijotham, Krishnamurthy [California Inst. of Technology (CalTech), Pasadena, CA (United States); Low, Steven [California Inst. of Technology (CalTech), Pasadena, CA (United States); Chertkov, Michael [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2015-07-21
The AC power flow equations underlie all operational aspects of power systems. They are solved routinely in operational practice using the Newton-Raphson method and its variants. These methods work well given a good initial “guess” for the solution, which is always available in normal system operations. However, with the increase in levels of intermittent generation, the assumption of a good initial guess always being available is no longer valid. In this paper, we solve this problem using the theory of monotone operators. We show that it is possible to compute (using an offline optimization) a “monotonicity domain” in the space of voltage phasors. Given this domain, there is a simple efficient algorithm that will either find a solution in the domain, or provably certify that no solutions exist in it. We validate the approach on several IEEE test cases and demonstrate that the offline optimization can be performed tractably and the computed “monotonicity domain” includes all practically relevant power flow solutions.
International Nuclear Information System (INIS)
Ka-Lin, Su; Yuan-Xi, Xie
2010-01-01
By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ordinary differential equation method is presented for solving the (2 + 1)-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the (2 + 1)-dimensional sine-Poisson equation are derived in a simple manner by this technique. (general)
Application of the trial equation method for solving some nonlinear ...
Indian Academy of Sciences (India)
the trial equation method. Also a more general trial equation method is proposed. Keywords. Trial equation method; KdV equation; K(m, n) equation; dual-power law; soliton solution. PACS Nos 02.30.Jr; 02.70.Wz; 04.20.Jb. 1. Introduction. Nonlinear phenomena exist in all the fields such as fluid mechanics, plasma physics, ...
DEFF Research Database (Denmark)
Hattel, Jesper; Hansen, Preben
1995-01-01
This paper presents a novel control volume based FD method for solving the equilibrium equations in terms of displacements, i.e. the generalized Navier equations. The method is based on the widely used cv-FDM solution of heat conduction and fluid flow problems involving a staggered grid formulation....... The resulting linear algebraic equations are solved by line-Gauss-Seidel....
Improved algorithm for solving nonlinear parabolized stability equations
International Nuclear Information System (INIS)
Zhao Lei; Zhang Cun-bo; Liu Jian-xin; Luo Ji-sheng
2016-01-01
Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. (paper)
Improved algorithm for solving nonlinear parabolized stability equations
Zhao, Lei; Zhang, Cun-bo; Liu, Jian-xin; Luo, Ji-sheng
2016-08-01
Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. Project supported by the National Natural Science Foundation of China (Grant Nos. 11332007 and 11402167).
Application of the trial equation method for solving some nonlinear ...
Indian Academy of Sciences (India)
In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the (, ) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.
Aguareles, M.
2014-06-01
In this paper we consider an oscillatory medium whose dynamics are modeled by the complex Ginzburg-Landau equation. In particular, we focus on n-armed spiral wave solutions of the complex Ginzburg-Landau equation in a disk of radius d with homogeneous Neumann boundary conditions. It is well-known that such solutions exist for small enough values of the twist parameter q and large enough values of d. We investigate the effect of boundaries on the rotational frequency of the spirals, which is an unknown of the problem uniquely determined by the parameters d and q. We show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak. We use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q. © 2014 Elsevier B.V. All rights reserved.
Method of mechanical quadratures for solving singular integral equations of various types
Sahakyan, A. V.; Amirjanyan, H. A.
2018-04-01
The method of mechanical quadratures is proposed as a common approach intended for solving the integral equations defined on finite intervals and containing Cauchy-type singular integrals. This method can be used to solve singular integral equations of the first and second kind, equations with generalized kernel, weakly singular equations, and integro-differential equations. The quadrature rules for several different integrals represented through the same coefficients are presented. This allows one to reduce the integral equations containing integrals of different types to a system of linear algebraic equations.
Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (3).
Murase, Kenya
2016-01-01
In this issue, simultaneous differential equations were introduced. These differential equations are often used in the field of medical physics. The methods for solving them were also introduced, which include Laplace transform and matrix methods. Some examples were also introduced, in which Laplace transform and matrix methods were applied to solving simultaneous differential equations derived from a three-compartment kinetic model for analyzing the glucose metabolism in tissues and Bloch equations for describing the behavior of the macroscopic magnetization in magnetic resonance imaging.In the next (final) issue, partial differential equations and various methods for solving them will be introduced together with some examples in medical physics.
The modified simple equation method for solving some fractional ...
Indian Academy of Sciences (India)
... in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinearfractional Klein–Gordon equation by using the modified simple equation method.
The modified simple equation method for solving some fractional ...
Indian Academy of Sciences (India)
Home; Journals; Pramana – Journal of Physics; Volume 87; Issue 1. The modified simple equation method for ... Nonlinear fractional differential equations are encountered in various fields of mathematics, physics, chemistry, biology, engineering and in numerous other applications. Exact solutions of these equations play a ...
Equations as Guides to Thinking and Problem Solving
Hewitt, Paul G.
2011-01-01
Science is the study of nature's rules. The most basic of these are the laws of physics, most of which are expressed in equation form. Physics equations show how concepts connect to one another. But does a study of these equations enhance student understanding? Not always, for too often in an introductory course students are tempted (or even…
About Ginzburg-Landau, and a bit on others
International Nuclear Information System (INIS)
Maksimov, Evgenii G
2011-01-01
This note is a brief history of how the theory of Ginzburg and Landau came to be. Early publications on the macroscopic theory of superconductivity are reviewed in detail. Discussions that the two co-authors had with their colleagues and between themselves are described. The 1952 review by V L Ginzburg is discussed, in which a number of well-defined requirements on the yet-to-be-developed microscopic theory of superconductivity were formulated, constituting what J Bardeen called the 'Ginzburg energy gap model'. (from the history of physics)
Nonlinear theory of deformable superconductors: Ginzburg-Landau description
Czech Academy of Sciences Publication Activity Database
Lipavský, Pavel; Morawetz, K.; Koláček, Jan; Brandt, E. H.
2008-01-01
Roč. 78, č. 17 (2008), 174516/1-174516/7 ISSN 1098-0121 R&D Projects: GA ČR GA202/08/0326; GA AV ČR IAA100100712; GA ČR(CZ) GA202/06/0040; GA AV ČR IAA1010404 Institutional research plan: CEZ:AV0Z10100521 Keywords : superconductivity * magneto-elastic effect * inhomogeneous superconductor Subject RIV: BM - Solid Matter Physics ; Magnetism Impact factor: 3.322, year: 2008
A Python Program for Solving Schro¨dinger's Equation in Undergraduate Physical Chemistry
Srnec, Matthew N.; Upadhyay, Shiv; Madura, Jeffry D.
2017-01-01
In undergraduate physical chemistry, Schrödinger's equation is solved for a variety of cases. In doing so, the energies and wave functions of the system can be interpreted to provide connections with the physical system being studied. Solving this equation by hand for a one-dimensional system is a manageable task, but it becomes time-consuming…
International Nuclear Information System (INIS)
Sharifi, M. J.; Adibi, A.
2000-01-01
In this paper, we have extended and completed our previous work, that was introducing a new method for finite differentiation. We show the applicability of the method for solving a wide variety of equations such as poisson, Laplace and Schrodinger. These equations are fundamental to the most semiconductor device simulators. In a section, we solve the Shordinger equation by this method in several cases including the problem of finding electron concentration profile in the channel of a HEMT. In another section, we solve the Poisson equation by this method, choosing the problem of SBD as an example. Finally we solve the Laplace equation in two dimensions and as an example, we focus on the VED. In this paper, we have shown that, the method can get stable and precise results in solving all of these problems. Also the programs which have been written based on this method become considerably faster, more clear, and more abstract
Ket-Bra entangled state method for solving master equation of finite-level system
Ren, Yi-Chong; Wang, Shu; Fan, Hong-Yi; Chen, Feng
2017-11-01
In this paper, we first introduce Ket-Bra entangled state method to solve master equation of finite-level system, which can convert master equation into Schrödinger-like equation and solve it with the mature methodology of Schrödinger equation. Then, several physical models include a radioactivity damped 2-level atom driven by classical field, a J- C model with cavity damping, a V-type qutrit under amplitude damping and N-qubits open Heisenberg chain have been solved with KBES method. Furthermore, the dynamic evolution and decoherence process of these models are investigated.
On an improved method for solving evolution equations of higher ...
African Journals Online (AJOL)
In this paper we introduce a new algebraic procedure to compute new classes of solutions of (1+1)-nonlinear partial differential equations (nPDEs) both of physical and technical relevance. The basic assumption is that the unknown solution(s) of the nPDE under consideration satisfy an ordinary differential equation (ODE) of ...
The modified simple equation method for solving some fractional ...
Indian Academy of Sciences (India)
2016-06-21
Jun 21, 2016 ... crucial role in the proper understanding of the qualitative features of many phenomena and processes in various areas of natural ... Keywords. Fractional differential equation; fractional complex transform; modified simple equation method; ... more frequently in various research and engineering applications ...
An algebraic method for solving Hartree-Fock-Roothaan equations
International Nuclear Information System (INIS)
Malbouisson, L.; Vianna, J.D.M.
1990-01-01
A mathematical method for studying of Hartree - Fock equations in LCAO approximation is presented. The method provides Hartree - Fock - Roothaan solutions for any occupancy scheme and does not use ordering rules of the usual iterative self-consistent procedures. Equations and applications for closed-shell configuration are considered explicitly
A highly accurate method to solve Fisher's equation
Indian Academy of Sciences (India)
2u. ∂x2 + u(1 − u). (2). Equation (2) may be transformed into an ordinary differential equation by substituting u = u(z) = u(x − ct). Kolmogorov et al [9] showed that with appropriate initial and. DOI: 10.1007/s12043-011-0243-8; ePublication: 28 ...
A Model for Solving the Maxwell Quasi Stationary Equations in a 3-Phase Electric Reduction Furnace
Directory of Open Access Journals (Sweden)
S. Ekrann
1982-10-01
Full Text Available A computer code has been developed for the approximate computation of electric and magnetic fields within an electric reduction furnace. The paper describes the numerical methods used to solve Maxwell's quasi-stationary equations, which are the governing equations for this problem. The equations are discretized by a staggered grid finite difference technique. The resulting algebraic equations are solved by iterating between computations of electric and magnetic quantities. This 'outer' iteration converges only when the skin depth is larger or of about the same magnitude as the linear dimensions of the computational domain. In solving for electric quantities with magnetic quantities being regarded as known, and vice versa, the central computational task is the solution of a Poisson equation for a scalar potential. These equations are solved by line successive overrelaxation combined with a rebalancing technique.
On supporting students' understanding of solving linear equation by using flowchart
Toyib, Muhamad; Kusmayadi, Tri Atmojo; Riyadi
2017-05-01
The aim of this study was to support 7th graders to gradually understand the concepts and procedures of solving linear equation. Thirty-two 7th graders of a Junior High School in Surakarta, Indonesia were involved in this study. Design research was used as the research approach to achieve the aim. A set of learning activities in solving linear equation with one unknown were designed based on Realistic Mathematics Education (RME) approach. The activities were started by playing LEGO to find a linear equation then solve the equation by using flowchart. The results indicate that using the realistic problems, playing LEGO could stimulate students to construct linear equation. Furthermore, Flowchart used to encourage students' reasoning and understanding on the concepts and procedures of solving linear equation with one unknown.
Directory of Open Access Journals (Sweden)
Veyis Turut
2013-01-01
Full Text Available Two tecHniques were implemented, the Adomian decomposition method (ADM and multivariate Padé approximation (MPA, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM, then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.
A new technique for solving the 1-D burgers equation
Directory of Open Access Journals (Sweden)
Yang Xiaojun
2017-01-01
Full Text Available In this paper, we address a new computational method, which is called the decomposition-Sumudu-like-integral-transform method, to handle the 1-D Burgers equation. The proposed method enables the efficient and accurate.
A novel approach for solving fractional Fisher equation using ...
Indian Academy of Sciences (India)
2u. ∂x2 + u(1 ... same equation occurs in logistic population growth models, flame propagation, neuro- physiology, autocatalytic ... transform is an iterative procedure for obtaining analytic Taylor series solution of ordinary or partial differential ...
Goldston, J. W.
This unit introduces analytic solutions of ordinary differential equations. The objective is to enable the student to decide whether a given function solves a given differential equation. Examples of problems from biology and chemistry are covered. Problem sets, quizzes, and a model exam are included, and answers to all items are provided. The…
Fractional differential equations solved by using Mellin transform
Butera, Salvatore; Di Paola, Mario
2014-01-01
In this paper, the solution of the multi-order differential equations, by using Mellin Transform, is proposed. It is shown that the problem related to the shift of the real part of the argument of the transformed function, arising when the Mellin integral operates on the fractional derivatives, may be overcame. Then, the solution may be found for any fractional differential equation involving multi-order fractional derivatives (or integrals). The solution is found in the Mellin domain, by sol...
DEFF Research Database (Denmark)
Reck, Kasper; Thomsen, Erik Vilain; Hansen, Ole
2011-01-01
. The solution of the mapped Helmholtz equation is found by solving an infinite series of Poisson equations using two dimensional Fourier series. The solution is entirely based on analytical expressions and is not mesh dependent. The analytical results are compared to a numerical (finite element method) solution......The scalar wave equation, or Helmholtz equation, describes within a certain approximation the electromagnetic field distribution in a given system. In this paper we show how to solve the Helmholtz equation in complex geometries using conformal mapping and the homotopy perturbation method...
A highly accurate method to solve Fisher's equation
Indian Academy of Sciences (India)
) for our discussion, one can apply the proposed method to solve the more general problem. ∂u. ∂t. = ∂2u. ∂x2 + F(u), where F is a real function. Therefore, we consider this problem for our numerical examples. Example 1. Consider the ...
New Numerical Treatment for Solving the KDV Equation
Directory of Open Access Journals (Sweden)
khalid ali
2017-01-01
Full Text Available In the present article, a numerical method is proposed for the numerical solution of the KdV equation by using collocation method with the modified exponential cubic B-spline. In this paper we convert the KdV equation to system of two equations. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms2L, ?L are computed. Three invariants of motion are predestined to determine the preservation properties of the problem, and the numerical scheme leads to careful and active results. Furthermore, interaction of two and three solitary waves is shown. These results show that the technique introduced here is easy to apply.
Solving the Linear 1D Thermoelasticity Equations with Pure Delay
Directory of Open Access Journals (Sweden)
Denys Ya. Khusainov
2015-01-01
Full Text Available We propose a system of partial differential equations with a single constant delay τ>0 describing the behavior of a one-dimensional thermoelastic solid occupying a bounded interval of R1. For an initial-boundary value problem associated with this system, we prove a well-posedness result in a certain topology under appropriate regularity conditions on the data. Further, we show the solution of our delayed model to converge to the solution of the classical equations of thermoelasticity as τ→0. Finally, we deduce an explicit solution representation for the delay problem.
Directory of Open Access Journals (Sweden)
Thomas Gomez
2018-04-01
Full Text Available Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods. Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numerical complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. This technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.
Fibonacci-regularization method for solving Cauchy integral equations of the first kind
Directory of Open Access Journals (Sweden)
Mohammad Ali Fariborzi Araghi
2017-09-01
Full Text Available In this paper, a novel scheme is proposed to solve the first kind Cauchy integral equation over a finite interval. For this purpose, the regularization method is considered. Then, the collocation method with Fibonacci base function is applied to solve the obtained second kind singular integral equation. Also, the error estimate of the proposed scheme is discussed. Finally, some sample Cauchy integral equations stem from the theory of airfoils in fluid mechanics are presented and solved to illustrate the importance and applicability of the given algorithm. The tables in the examples show the efficiency of the method.
Solving Multi-variate Polynomial Equations in a Finite Field
2013-06-01
hardware to encrypt and decrypt messages. Many of the AES predecessors use this Feistel structure (i.e. DES, Lucifer , Blowfish). However, AES does not...However, then it is very effective . The interesting aspect about the agreeing algorithm is that it can gain momentum to solve the system once RHSs are...columns from Lh can now be removed. This can create a ‘cascade effect ’ on the system and the system quickly reduces its size and complexity. Agreeing
Projection-iteration methods for solving nonlinear operator equations
International Nuclear Information System (INIS)
Nguyen Minh Chuong; Tran thi Lan Anh; Tran Quoc Binh
1989-09-01
In this paper, the authors investigate a nonlinear operator equation in uniformly convex Banach spaces as in metric spaces by using stationary and nonstationary generalized projection-iteration methods. Convergence theorems in the strong and weak sense were established. (author). 7 refs
A novel approach for solving fractional Fisher equation using ...
Indian Academy of Sciences (India)
In the present paper, an analytic solution of nonlinear fractional Fisher equation is deduced with the help of the powerful differential transform method (DTM). ... Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, P.C. 44891-63157, Rudsar-Vajargah, Iran ...
The modified simple equation method for solving some fractional ...
Indian Academy of Sciences (India)
2016-06-21
Jun 21, 2016 ... crucial role in the proper understanding of the qualitative features of many phenomena and processes in various areas of natural science. Thus, many effective and powerful methods have ... ified trial equation method [18] and so on [19] have been proposed to obtain exact solutions of fractional differential ...
How fast the Laplace equation was solved in 1995
Botta, E.F.F.; Dekker, K.; Notay, Y.; Ploeg, A. van der; Vuik, C.; Wubs, F.W.
On the occasion of the third centenary of the appointment of Johann Bernoulli at the University of Groningen, a number of linear systems solvers for some Laplace-like equations have been compared during a one-day workshop. CPU times of several advanced solvers measured on the same computer (an
Rebello, Carina M.; Rebello, N. Sanjay
2012-02-01
Previous studies have focused on the resources that students activate and utilize while solving a given physics problem. However, few studies explore how students relate a given resource such as an equation, to various types of physics problems and contexts and how they ascertain the meaning and applicability of that resource. We explore how students view physics equations, derive meaning from those equations, and use those equations in physics problem solving. We adapt Dubinsky and McDonald's description of APOS (action-process-object-schema) theory of learning in mathematics, to construct a theoretical framework that describes how students interpret and use equations in physics in terms of actions, processes, objects, and schemas. This framework provides a lens for understanding how students construct their understanding of physics concepts and their relation to equations. We highlight how APOS theory can be operationalized to serve as a lens for studying the use of mathematics in physics problem solving.
Numerical treatments for solving nonlinear mixed integral equation
Directory of Open Access Journals (Sweden)
M.A. Abdou
2016-12-01
Full Text Available We consider a mixed type of nonlinear integral equation (MNLIE of the second kind in the space C[0,T]×L2(Ω,T<1. The Volterra integral terms (VITs are considered in time with continuous kernels, while the Fredholm integral term (FIT is considered in position with singular general kernel. Using the quadratic method and separation of variables method, we obtain a nonlinear system of Fredholm integral equations (NLSFIEs with singular kernel. A Toeplitz matrix method, in each case, is then used to obtain a nonlinear algebraic system. Numerical results are calculated when the kernels take a logarithmic form or Carleman function. Moreover, the error estimates, in each case, are then computed.
Extension of the homotopy pertubation method for solving nonlinear differential-difference equations
Energy Technology Data Exchange (ETDEWEB)
Mousa, Mohamed Medhat [Benha Univ. (Egypt). Benha High Inst. of Technology; Al-Farabi Kazakh National Univ., Almaty (Kazakhstan); Kaltayev, Aidarkan [Al-Farabi Kazakh National Univ., Almaty (Kazakhstan); Bulut, Hasan [Firat Univ., Elazig (Turkey). Dept. of Mathematics
2010-12-15
In this paper, we have extended the homotopy perturbation method (HPM) to find approximate analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Korteweg-de Vries (mKdV) lattice equation and the discretized nonlinear Schroedinger equation are taken as examples to demonstrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons are made between the results of the presented method and exact solutions. The obtained results reveal that the HPM is a very effective and convenient tool for solving such kind of equations. (orig.)
Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (4).
Murase, Kenya
2016-01-01
Partial differential equations are often used in the field of medical physics. In this (final) issue, the methods for solving the partial differential equations were introduced, which include separation of variables, integral transform (Fourier and Fourier-sine transforms), Green's function, and series expansion methods. Some examples were also introduced, in which the integral transform and Green's function methods were applied to solving Pennes' bioheat transfer equation and the Fourier series expansion method was applied to Navier-Stokes equation for analyzing the wall shear stress in blood vessels.Finally, the author hopes that this series will be helpful for people who engage in medical physics.
One dimensional code to solve multigroup kinetic equations
International Nuclear Information System (INIS)
Alcantara, H.G. de; Prati, A.; Rosa, M.A.P.; Nair, R.P.K.
1985-01-01
It is described a computer program for the numerical solution of neutron kinetic equations in the multigroup theory for one dimensional medium. The spatial dependence is discretized by finite differences. The time integration is obtained by the method of ponderated residuals and iterative solution. It is examined one method of convergence acceleration. The program studies the reactivity feedback by the variation of temperature or density. It is simulated the simplified model of heat extraction. (M.C.K.) [pt
Algorithm for solving polynomial algebraic Riccati equations and its application
Czech Academy of Sciences Publication Activity Database
Augusta, Petr; Augustová, Petra
2012-01-01
Roč. 1, č. 4 (2012), s. 237-242 ISSN 2223-7038 R&D Projects: GA ČR GPP103/12/P494 Institutional support: RVO:67985556 Keywords : Numerical algorithms * algebraic Riccati equation * spatially distributed systems * optimal control Subject RIV: BC - Control Systems Theory http://lib.physcon.ru/doc?id=8b4876d6a57d
Equilibrium of charges and differential equations solved by polynomials
International Nuclear Information System (INIS)
Loutsenko, Igor
2004-01-01
We study limits of particular importance of the bilinear hypergeometric equation introduced in a previous paper. As part of this study, we examine connections between the rationality of certain indefinite integrals and the equilibrium of Coulomb charges in the complex plane (or point vortices in two-dimensional hydrodynamics). Relationships with integrable models which are generalizations of the Calogero-Moser systems are also discussed
Solving singular convolution equations using the inverse fast Fourier transform
Czech Academy of Sciences Publication Activity Database
Krajník, E.; Montesinos, V.; Zizler, P.; Zizler, Václav
2012-01-01
Roč. 57, č. 5 (2012), s. 543-550 ISSN 0862-7940 R&D Projects: GA AV ČR IAA100190901 Institutional research plan: CEZ:AV0Z10190503 Keywords : singular convolution equations * fast Fourier transform * tempered distribution Subject RIV: BA - General Mathematics Impact factor: 0.222, year: 2012 http://www.springerlink.com/content/m8437t3563214048/
Directly Solving Special Second Order Delay Differential Equations Using Runge-Kutta-Nyström Method
Directory of Open Access Journals (Sweden)
M. Mechee
2013-01-01
Full Text Available Runge-Kutta-Nyström (RKN method is adapted for solving the special second order delay differential equations (DDEs. The stability polynomial is obtained when this method is used for solving linear second order delay differential equation. A standard set of test problems is solved using the method together with a cubic interpolation for evaluating the delay terms. The same set of problems is reduced to a system of first order delay differential equations and then solved using the existing Runge-Kutta (RK method. Numerical results show that the RKN method is more efficient in terms of accuracy and computational time when compared to RK method. The methods are applied to a well-known problem involving delay differential equations, that is, the Mathieu problem. The numerical comparison shows that both methods are in a good agreement.
Variational iteration method for solving the multi-pantograph delay equation
International Nuclear Information System (INIS)
Yu Zhanhua
2008-01-01
In this Letter, the variational iteration method is applied to solve the multi-pantograph delay equation. Sufficient conditions are given to assure the convergence of the method. Examples show that the method is effective
Application of Trotter approximation for solving time dependent neutron transport equation
International Nuclear Information System (INIS)
Stancic, V.
1987-01-01
A method is proposed to solve multigroup time dependent neutron transport equation with arbitrary scattering anisotropy. The recurrence relation thus obtained is simple, numerically stable and especially suitable for treatment of complicated geometries. (author)
Solving differential–algebraic equation systems by means of index reduction methodology
DEFF Research Database (Denmark)
Sørensen, Kim; Houbak, Niels; Condra, Thomas
2006-01-01
of a number of differential equations and algebraic equations — a so called DAE system. Two of the DAE systems are of index 1 and they can be solved by means of standard DAE-solvers. For the actual application, the equation systems are integrated by means of MATLAB’s solver: ode23t, that solves moderately......, it is shown how the equation system, by means of an index reduction methodology, can be reduced to a system of ordinary differential equations — ODEs....... stiff ODEs and index 1 DAEs by means of the trapezoidal rule. The last sub-model that models the boilers steam drum consist of two differential and three algebraic equations. The index of this model is greater than 1, which means that ode23t cannot integrate this equation system. In this paper...
Solving differential-algebraic equation systems by means of index reduction methodology
DEFF Research Database (Denmark)
Sørensen, Kim; Houbak, Niels; Condra, Thomas Joseph
2006-01-01
of a number of differential equations and algebraic equations - a so called DAE system. Two of the DAE systems are of index 1 and they can be solved by means of standard DAE-solvers. For the actual application, the equation systems are integrated by means of MATLAB’s solver: ode23t, that solves moderately......, it is shown how the equation system, by means of an index reduction methodology, can be reduced to a system of Ordinary- Differential-Equations - ODE’s....... stiff ODE’s and index 1 DAE’s by means of the trapezoidal rule. The last sub-model that models the boilers steam drum consist of two differential and three algebraic equations. The index of this model is greater than 1, which means that ode23t cannot integrate this equation system. In this paper...
Ghanbari, Behzad
2014-01-01
We aim to study the convergence of the homotopy analysis method (HAM in short) for solving special nonlinear Volterra-Fredholm integrodifferential equations. The sufficient condition for the convergence of the method is briefly addressed. Some illustrative examples are also presented to demonstrate the validity and applicability of the technique. Comparison of the obtained results HAM with exact solution shows that the method is reliable and capable of providing analytic treatment for solving such equations.
Directory of Open Access Journals (Sweden)
Behzad Ghanbari
2014-01-01
Full Text Available We aim to study the convergence of the homotopy analysis method (HAM in short for solving special nonlinear Volterra-Fredholm integrodifferential equations. The sufficient condition for the convergence of the method is briefly addressed. Some illustrative examples are also presented to demonstrate the validity and applicability of the technique. Comparison of the obtained results HAM with exact solution shows that the method is reliable and capable of providing analytic treatment for solving such equations.
Directory of Open Access Journals (Sweden)
Mohammad Almousa
2013-01-01
Full Text Available The aim of this study is to present the use of a semi analytical method called the optimal homotopy asymptotic method (OHAM for solving the linear Fredholm integral equations of the first kind. Three examples are discussed to show the ability of the method to solve the linear Fredholm integral equations of the first kind. The results indicated that the method is very effective and simple.
An Algorithm for Solving the Absolute Value Equation
Czech Academy of Sciences Publication Activity Database
Rohn, Jiří
2009-01-01
Roč. 18, - (2009), s. 589-599 E-ISSN 1081-3810 R&D Projects: GA ČR GA201/09/1957; GA ČR GC201/08/J020 Institutional research plan: CEZ:AV0Z10300504 Keywords : absolute value equation * algorithm * regularity * singularity * theorem of the alternatives Subject RIV: BA - General Mathematics Impact factor: 0.892, year: 2009 http://www.math.technion.ac.il/iic/ ela / ela -articles/articles/vol18_pp589-599.pdf
On Solving Systems of Equations by Successive Reduction Using 2×2 Matrices
Carley, Holly
2014-01-01
Usually a student learns to solve a system of linear equations in two ways: "substitution" and "elimination." While the two methods will of course lead to the same answer they are considered different because the thinking process is different. In this paper the author solves a system in these two ways to demonstrate the…
Solving the high energy evolution equation including running coupling corrections
International Nuclear Information System (INIS)
Albacete, Javier L.; Kovchegov, Yuri V.
2007-01-01
We study the solution of the nonlinear Balitsky-Kovchegov evolution equation with the recently calculated running coupling corrections [I. I. Balitsky, Phys. Rev. D 75, 014001 (2007). and Y. Kovchegov and H. Weigert, Nucl. Phys. A784, 188 (2007).]. Performing a numerical solution we confirm the earlier result of Albacete et al. [Phys. Rev. D 71, 014003 (2005).] (obtained by exploring several possible scales for the running coupling) that the high energy evolution with the running coupling leads to a universal scaling behavior for the dipole-nucleus scattering amplitude, which is independent of the initial conditions. It is important to stress that the running coupling corrections calculated recently significantly change the shape of the scaling function as compared to the fixed coupling case, in particular, leading to a considerable increase in the anomalous dimension and to a slow-down of the evolution with rapidity. We then concentrate on elucidating the differences between the two recent calculations of the running coupling corrections. We explain that the difference is due to an extra contribution to the evolution kernel, referred to as the subtraction term, which arises when running coupling corrections are included. These subtraction terms were neglected in both recent calculations. We evaluate numerically the subtraction terms for both calculations, and demonstrate that when the subtraction terms are added back to the evolution kernels obtained in the two works the resulting dipole amplitudes agree with each other. We then use the complete running coupling kernel including the subtraction term to find the numerical solution of the resulting full nonlinear evolution equation with the running coupling corrections. Again the scaling regime is recovered at very large rapidity with the scaling function unaltered by the subtraction term
On the Efficiency of Algorithms for Solving Hartree–Fock and Kohn–Sham Response Equations
DEFF Research Database (Denmark)
Kauczor, Joanna; Jørgensen, Poul; Norman, Patrick
2011-01-01
The response equations as occurring in the Hartree–Fock, multiconfigurational self-consistent field, and Kohn–Sham density functional theory have identical matrix structures. The algorithms that are used for solving these equations are discussed, and new algorithms are proposed where trial vectors...
The H-N method for solving linear transport equation: theory and application
International Nuclear Information System (INIS)
Kaskas, A.; Gulecyuz, M.C.; Tezcan, C.
2002-01-01
The system of singular integral equation which is obtained from the integro-differential form of the linear transport equation as a result of Placzec lemma is solved. Application are given using the exit distributions and the infinite medium Green's function. The same theoretical results are also obtained with the use of the singular eigenfunction of the method of elementary solutions
Directory of Open Access Journals (Sweden)
Ai-Min Yang
2014-01-01
Full Text Available We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.
Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets
Directory of Open Access Journals (Sweden)
Ai-Min Yang
2013-01-01
Full Text Available We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.
Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method
Directory of Open Access Journals (Sweden)
Shao-Hong Yan
2014-01-01
Full Text Available The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.
Didis, Makbule Gozde; Erbas, Ayhan Kursat
2015-01-01
This study attempts to investigate the performance of tenth-grade students in solving quadratic equations with one unknown, using symbolic equation and word-problem representations. The participants were 217 tenth-grade students, from three different public high schools. Data was collected through an open-ended questionnaire comprising eight…
New approach to solve fully fuzzy system of linear equations using ...
Indian Academy of Sciences (India)
This paper proposes two new methods to solve fully fuzzy system of linear equations. The fuzzy system has been converted to a crisp system of linear equations by using single and double parametric form of fuzzy numbers to obtain the non-negative solution. Double parametric form of fuzzy numbers is defined and applied ...
New approach to solve fully fuzzy system of linear equations using ...
Indian Academy of Sciences (India)
Abstract. This paper proposes two new methods to solve fully fuzzy system of linear equations. The fuzzy system has been converted to a crisp system of linear equations by using single and double parametric form of fuzzy numbers to obtain the non-negative solution. Double parametric form of fuzzy numbers is defined and.
Trial equation method for solving the generalized Fisher equation with variable coefficients
Energy Technology Data Exchange (ETDEWEB)
Triki, Houria [Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba (Algeria); Wazwaz, Abdul-Majid, E-mail: wazwaz@sxu.edu [Department of Mathematics, Saint Xavier University, Chicago, IL 60655 (United States)
2016-03-22
We investigate a generalized Fisher equation with temporally varying coefficients, describing the dynamics of a field in inhomogeneous media. A class of exact soliton solutions of this equation is presented, and some of which are derived for the first time. The trial equation method is applied to obtain these soliton solutions. The constraint conditions for the existence of these solutions are also exhibited.
Solving momentum-space integral equations for quarkonia spectra with confining potentials
International Nuclear Information System (INIS)
Eyre, D.; Vary, J.P.
1986-01-01
Singular integral equations for quarkonia (qq-bar) spectra are solved in momentum space for different choices of confining potentials by introducing a regularization procedure. The method is sufficiently general to treat nonlocal potentials and combinations of singular potentials. Through nonrelativistic model applications we demonstrate the stability and accuracy of the method. The method works in all partial waves. A first-order correction to the eigenenergies brings calculated results for soluble model problems into remarkable agreement with exact results. Extensions of the method to solve the nonrelativistic spectra of three-quark systems and to solve the relativistic Bethe-Salpeter equation are discussed
International Nuclear Information System (INIS)
Koleshko, S.B.
1989-01-01
A three-parametric set of difference schemes is suggested to solve Navier-Stokes equations with the use of the relaxation form of the continuity equation. The initial equations are stated for time increments. Use is made of splitting the operator into one-dimensional forms that reduce calculations to scalar factorizations. Calculated results for steady- and unsteady-state flows in a cavity are presented
The change of the brain activation patterns as children learn algebra equation solving
Qin, Yulin; Carter, Cameron S.; Silk, Eli M.; Stenger, V. Andrew; Fissell, Kate; Goode, Adam; Anderson, John R.
2004-04-01
In a brain imaging study of children learning algebra, it is shown that the same regions are active in children solving equations as are active in experienced adults solving equations. As with adults, practice in symbol manipulation produces a reduced activation in prefrontal cortex area. However, unlike adults, practice seems also to produce a decrease in a parietal area that is holding an image of the equation. This finding suggests that adolescents' brain responses are more plastic and change more with practice. These results are integrated in a cognitive model that predicts both the behavioral and brain imaging results.
Effective methods of solving of model equations of certain class of thermal systems
International Nuclear Information System (INIS)
Lach, J.
1985-01-01
A number of topics connected with solving of model equations of certain class of thermal systems by the method of successive approximations is touched. A system of partial differential equations of the first degree, appearing most frequently in practical applications of heat and mass transfer theory is reduced to an equivalent system of Volterra integral equations of the second kind. Among a few sample applications the thermal processes appearing in the fuel channel of nuclear reactor are solved. The theoretical analysis is illustrated by the results of numerical calculations given in tables and diagrams. 111 refs., 17 figs., 16 tabs. (author)
Energy Technology Data Exchange (ETDEWEB)
Li,Jing; Tu, Xuemin
2008-12-10
A variant of balancing domain decomposition method by constraints (BDDC) is proposed for solving a class of indefinite system of linear equations, which arises from the finite element discretization of the Helmholtz equation of time-harmonic wave propagation in a bounded interior domain. The proposed BDDC algorithm is closely related to the dual-primal finite element tearing and interconnecting algorithm for solving Helmholtz equations (FETI-DPH). Under the condition that the diameters of the subdomains are small enough, the rate of convergence is established which depends polylogarithmically on the dimension of the individual subdomain problems and which improves with the decrease of the subdomain diameters. These results are supported by numerical experiments of solving a Helmholtz equation on a two-dimensional square domain.
Gourgoulhon, Eric
2011-04-01
Numerical relativity is one of the major fields of contemporary general relativity and is developing continually. Yet three years ago, no textbook was available on this subject. The first textbook devoted to numerical relativity, by Alcubierre, appeared in 2008 [1] (cf the CQG review [2]). Now comes the second book, by Baumgarte and Shapiro, two well known players in the field. Inevitably, the two books have some common aspects (otherwise they would not deal with the same topic!). For instance the titles of the first four chapters of Baumgarte and Shapiro are very similar to those of Alcubierre. This arises from some logic inherent to the subject: chapter 1 recaps basic GR, chapter 2 introduces the 3+1 formalism, chapter 3 focuses on the initial data and chapter 4 on the choice of coordinates for the evolution. But there are also many differences between the two books, which actually make them complementary. At first glance the differences are the size (720 pages for Baumgarte and Shapiro vs 464 pages for Alcubierre) and the colour figures in Baumgarte and Shapiro. Regarding the content, Baumgarte and Shapiro address many topics which are not present in Alcubierre's book, such as magnetohydrodynamics, radiative transfer, collisionless matter, spectral methods, rotating stars and post-Newtonian approximation. The main difference regards binary systems: virtually absent from Alcubierre's book (except for binary black hole initial data), they occupy not less than five chapters in Baumgarte and Shapiro's book. In contrast, gravitational wave extraction, various hyperbolic formulations of Einstein's equations and the high-resolution shock-capturing schemes are treated in more depth by Alcubierre. In the first four chapters mentioned above, some distinctive features of Baumgarte and Shapiro's book are the beautiful treatment of Oppenheimer-Snyder collapse in chapter 1, the analogy with Maxwell's equations when discussing the constraints and the evolution equations in
Solving the dirac equation with nonlocal potential by imaginary time step method
International Nuclear Information System (INIS)
Zhang Ying; Liang Haozhao; Meng Jie
2009-01-01
The imaginary time step (ITS) method is applied to solve the Dirac equation with the nonlocal potential in coordinate space by the ITS evolution for the corresponding Schroedinger-like equation for the upper component. It is demonstrated that the ITS evolution can be equivalently performed for the Schroedinger-like equation with or without localization. The latter algorithm is recommended in the application for the reason of simplicity and efficiency. The feasibility and reliability of this algorithm are also illustrated by taking the nucleus 16 O as an example, where the same results as the shooting method for the Dirac equation with localized effective potentials are obtained. (authors)
DEFF Research Database (Denmark)
Hejlesen, Mads Mølholm
A regularisation method for solving the Poisson equation using Green’s functions is presented.The method is shown to obtain a convergence rate which corresponds to the design of the regularised Green’s function and a spectral-like convergence rate is obtained using a spectrally ideal regularisation....... It is shown that the regularised Poisson solver can be extended to handle mixed periodic and free-space boundary conditions. This is done by solving the equation spectrally in the periodic directions which yields a modified Helmholtz equation for the free-space directions which in turn is solved by deriving...... the appropriate regularised Green’s functions. Using an analogy to the particle-particle particle-mesh method, a framework for calculating multi-resolution solutions using local refinement patches is presented. The regularised Poisson solver is shown to maintain a high order converging solution for different...
Directory of Open Access Journals (Sweden)
N.M. Ghasem
2003-12-01
Full Text Available In this paper, the simulink block diagram is used to solve a model consists of a set of ordinary differential and algebraic equations to control the temperature inside a simple stirred tank heater. The flexibility of simulink block diagram gives students a better understanding of the control systems. The simulink also allows solution of mathematical models and easy visualization of the system variables. A polyethylene fluidized bed reactor is considered as an industrial example and the effect of the Proportional, Integral and Derivative control policy is presented for comparison.
Solving inverse problems for biological models using the collage method for differential equations.
Capasso, V; Kunze, H E; La Torre, D; Vrscay, E R
2013-07-01
In the first part of this paper we show how inverse problems for differential equations can be solved using the so-called collage method. Inverse problems can be solved by minimizing the collage distance in an appropriate metric space. We then provide several numerical examples in mathematical biology. We consider applications of this approach to the following areas: population dynamics, mRNA and protein concentration, bacteria and amoeba cells interaction, tumor growth.
Analytic solutions to a family of boundary-value problems for Ginsburg-Landau type equations
Vassilev, V. M.; Dantchev, D. M.; Djondjorov, P. A.
2017-10-01
We consider a two-parameter family of nonlinear ordinary differential equations describing the behavior of a critical thermodynamic system, e.g., a binary liquid mixture, of film geometry within the framework of the Ginzburg-Landau theory by means of the order-parameter. We focus on the case in which the confining surfaces are strongly adsorbing but prefer different components of the mixture, i.e., the order-parameter tends to infinity at one of the boundaries and to minus infinity at the other one. We assume that the boundaries of the system are positioned at a finite distance from each other and give analytic solutions to the corresponding boundary-value problems in terms of Weierstrass and Jacobi elliptic functions.
The application of the Galerkin method to solving PIES for Laplace's equation
Bołtuć, Agnieszka; Zieniuk, Eugeniusz
2016-06-01
The paper presents the application of the Galerkin method to solving the parametric integral equation system (PIES) on the example of Laplace's equation. The main aim of the paper is the analysis of the effectiveness of two methods for PIES solving: the collocation method and the Galerkin method. Researches were performed on two examples with analytical solutions. Tests concern mainly the accuracy of obtained numerical solutions and their stability. For both analyzed methods calculations were made with the various number of expressions in the approximation series, whilst in the collocation method two variants of the arrangement of collocation points were considered. We also compared the complexity of both methods using the execution time.
Nonlinear evolution equations and solving algebraic systems: the importance of computer algebra
International Nuclear Information System (INIS)
Gerdt, V.P.; Kostov, N.A.
1989-01-01
In the present paper we study the application of computer algebra to solve the nonlinear polynomial systems which arise in investigation of nonlinear evolution equations. We consider several systems which are obtained in classification of integrable nonlinear evolution equations with uniform rank. Other polynomial systems are related with the finding of algebraic curves for finite-gap elliptic potentials of Lame type and generalizations. All systems under consideration are solved using the method based on construction of the Groebner basis for corresponding polynomial ideals. The computations have been carried out using computer algebra systems. 20 refs
A homotopy method for solving Riccati equations on a shared memory parallel computer
International Nuclear Information System (INIS)
Zigic, D.; Watson, L.T.; Collins, E.G. Jr.; Davis, L.D.
1993-01-01
Although there are numerous algorithms for solving Riccati equations, there still remains a need for algorithms which can operate efficiently on large problems and on parallel machines. This paper gives a new homotopy-based algorithm for solving Riccati equations on a shared memory parallel computer. The central part of the algorithm is the computation of the kernel of the Jacobian matrix, which is essential for the corrector iterations along the homotopy zero curve. Using a Schur decomposition the tensor product structure of various matrices can be efficiently exploited. The algorithm allows for efficient parallelization on shared memory machines
A predictor-corrector scheme for solving the Volterra integral equation
Al Jarro, Ahmed
2011-08-01
The occurrence of late time instabilities is a common problem of almost all time marching methods developed for solving time domain integral equations. Implicit marching algorithms are now considered stable with various efforts that have been developed for removing low and high frequency instabilities. On the other hand, literature on stabilizing explicit schemes, which might be considered more efficient since they do not require a matrix inversion at each time step, is practically non-existent. In this work, a stable but still explicit predictor-corrector scheme is proposed for solving the Volterra integral equation and its efficacy is verified numerically. © 2011 IEEE.
Chedjou, Jean Chamberlain; Kyamakya, Kyandoghere
2015-04-01
This paper develops and validates a comprehensive and universally applicable computational concept for solving nonlinear differential equations (NDEs) through a neurocomputing concept based on cellular neural networks (CNNs). High-precision, stability, convergence, and lowest-possible memory requirements are ensured by the CNN processor architecture. A significant challenge solved in this paper is that all these cited computing features are ensured in all system-states (regular or chaotic ones) and in all bifurcation conditions that may be experienced by NDEs.One particular quintessence of this paper is to develop and demonstrate a solver concept that shows and ensures that CNN processors (realized either in hardware or in software) are universal solvers of NDE models. The solving logic or algorithm of given NDEs (possible examples are: Duffing, Mathieu, Van der Pol, Jerk, Chua, Rössler, Lorenz, Burgers, and the transport equations) through a CNN processor system is provided by a set of templates that are computed by our comprehensive templates calculation technique that we call nonlinear adaptive optimization. This paper is therefore a significant contribution and represents a cutting-edge real-time computational engineering approach, especially while considering the various scientific and engineering applications of this ultrafast, energy-and-memory-efficient, and high-precise NDE solver concept. For illustration purposes, three NDE models are demonstratively solved, and related CNN templates are derived and used: the periodically excited Duffing equation, the Mathieu equation, and the transport equation.
Directory of Open Access Journals (Sweden)
Enrique Castillo
2016-01-01
Full Text Available We first show that monomial ratio equations are not only very common in Physics and Engineering, but the natural type of equations in many practical problems. More precisely, in the case of models involving scale variables if the used formulas are not of this type they are not physically valid. The consequence is that when estimating the model parameters we are faced with systems of monomial ratio equations that are nonlinear and difficult to solve. In this paper, we provide an original algorithm to obtain the unique solutions of systems of equations made of linear combinations of monomial ratios whose coefficient matrix has a proper null space with low dimension that permits solving the problem in a simple way. Finally, we illustrate the proposed methods by their application to two practical problems from the hydraulic and structural fields.
Wu, Zedong
2018-04-05
Numerical simulation of the acoustic wave equation in either isotropic or anisotropic media is crucial to seismic modeling, imaging and inversion. Actually, it represents the core computation cost of these highly advanced seismic processing methods. However, the conventional finite-difference method suffers from severe numerical dispersion errors and S-wave artifacts when solving the acoustic wave equation for anisotropic media. We propose a method to obtain the finite-difference coefficients by comparing its numerical dispersion with the exact form. We find the optimal finite difference coefficients that share the dispersion characteristics of the exact equation with minimal dispersion error. The method is extended to solve the acoustic wave equation in transversely isotropic (TI) media without S-wave artifacts. Numerical examples show that the method is is highly accurate and efficient.
Intuitive physics knowledge, physics problem solving and the role of mathematical equations
Directory of Open Access Journals (Sweden)
Laura Buteler
2012-09-01
Full Text Available The present work explores the role that mathematical equations play in modifying students’ physical intuition (diSessa, 1993. The work is carried out assuming that students achieve a great deal of the refinement in their physical intuitions during problem solving (Sherin, 2006. The study is guided by the question of how the use of mathematical equations contributes to this refinement. The authors aim at expanding on Sherin´s (2006 hypothesis, suggesting a more bounding relation between physical intuitions and mathematics. In this scenario, intuitions play a more compelling role in “deciding” which equations are acceptable and which are not. Our hypothesis is constructed on the basis of three cases: the first published by Sherin (2006 and two more from registries of our own. The three cases are compared and analyzed in relation to the role of mathematical equations in refining – or not – the intuitive knowledge students bring to play during problem solving.
International Nuclear Information System (INIS)
Samba, G.
1985-04-01
It is often desirable to solve the two dimensional multigroup transport equation for (r-z) geometries directly given by hydrodynamic calculations. Usually, only Monte-Carlo codes are able to compute α or k eigenvalues on such geometries. Most deterministic codes use an orthogonal mesh or restrict the mesh to a regular triangular grid. Other methods were developed in C.E.A. and Los Alamos but do not solve the problem of sliding between two Lagrangian blocks. Thus, we have developed a production code which solves these problems and is able to get α or k eigenvalues with a good accuracy for such geometries
Auger, J C
2003-01-01
The multiple scattering problem can be solved using various analytical techniques. One of these techniques, the T-matrix formalism, is at the present time generally solved using iterative algorithms, because the initially proposed recursive algorithms appeared to be numerically unstable. We present here a new set of recursive relations to solve the multiple scattering equation, and discuss their range of application. In order to validate this new formalism, we compare numerical results for various complex systems with the Generalized Multi-particle Mie solution. We show that the results obtained with the recursive method are in very good agreement with those given by iterative techniques.
Cortes, Adriano Mauricio
2014-01-01
In this paper we present PetIGA, a high-performance implementation of Isogeometric Analysis built on top of PETSc. We show its use in solving nonlinear and time-dependent problems, such as phase-field models, by taking advantage of the high-continuity of the basis functions granted by the isogeometric framework. In this work, we focus on the Cahn-Hilliard equation and the phase-field crystal equation.
International Nuclear Information System (INIS)
Hudson, S.R.
2010-01-01
A method for approximately solving magnetic differential equations is described. The approach is to include a small diffusion term to the equation, which regularizes the linear operator to be inverted. The extra term allows a 'source-correction' term to be defined, which is generally required in order to satisfy the solvability conditions. The approach is described in the context of computing the pressure and parallel currents in the iterative approach for computing magnetohydrodynamic equilibria.
Waleed K. Ahmed
2013-01-01
The present paper demonstrates the route used for solving differential equations for the engineering applications at UAEU. Usually students at the Engineering Requirements Unit (ERU) stage of the Faculty of Engineering at the UAEU must enroll in a course of Differential Equations and Engineering Applications (MATH 2210) as a prerequisite for the subsequent stages of their study. Mainly, one of the objectives of this course is that the students practice MATLAB software package during the cours...
The Operator Method for Solving The Fractional Fokker-Planck Equation
International Nuclear Information System (INIS)
Zahran, M.A.; Abdou, M.A.
2002-01-01
The operator method has been used to solve the fractional Fokker-Planck equation which was recently formulated as a model for the anomalous transport process. Two classes of special interest of fractional F-P equations coming from plasma physics and charged particle transport problem has been considered. It is shown that the mean square-displacement 2 (t) > satisfy the universal power law characterized the anomalous time evolution e.i. 2 (t) >t γ , 0 < γ < 1
New generalized phase shift approach to solve the Helmholtz acoustic wave equation
Abeykoon, Sameera K. (Nee Rajapakshe)
2008-10-01
We have developed and given some proof of concept applications of a new method of solving the Helmholtz wave equation in order to facilitate the exploration of oil and gas. The approach is based on a new way to generalize the "one-way" wave equation, and to impose correct boundary conditions. The full two-way nature of the Helmholtz equation is considered, but converted into a pseudo "one-way" form with a generalized phase shift structure for propagation in the depth z. Two coupled first order partial differential equations in the depth variable z are obtained from the Helmholtz wave equation. Our approach makes use of very simple, standard ideas from differential equations and early ideas on the non-iterative solution of the Lippmann-Schwinger equation in quantum scattering. In addition, a judicious choice of operator splitting is introduced to ensure that only explicit solution techniques are required. This avoids the need for numerical matrix inversions. The initial conditions are more challenging due to the need to ensure that the solution satisfies proper boundary conditions associated with the waves traveling in two directions. This difficulty is resolved by solving the Lippmann-Schwinger integral equation in an explicit, non-iterative fashion. It is solved by essentially "factoring out" the physical boundary conditions, thereby converting the inhomogeneous Lippmann-Schwinger integral equation of the second kind into a Volterra integral equation of the second kind. Due to the special structure of the kernel, which is a consequence of the causal nature of the Green's function in the Lippmann-Schwinger equation, this turns out to be extremely efficient. The coupled first order differential equations will be solved using the "modified Cayley method" developed in Kouri's group some years ago. The Feshbach projection operator technique is used for constructing a solution that is stable with respect to "evanescent" or "non-propagating" waves. This method is
Diffuse optical tomography through solving a system of quadratic equations: theory and simulations.
Kanmani, B; Vasu, R M
2006-02-21
This paper discusses the iterative solution of the nonlinear problem of optical tomography. In the established forward model-based iterative image reconstruction (MOBIIR) method a linear perturbation equation containing the first derivative of the forward operator is solved to obtain the update vector for the optical properties. In MOBIIR, the perturbation equation is updated by recomputing the first derivative after each update of the optical properties. In the method presented here a nonlinear perturbation equation, containing terms up to the second derivative, is used to iteratively solve for the optical property updates. Through this modification, reconstructions with reasonable contrast recovery and accuracy are obtained without the need for updating the perturbation equation and therefore eliminating the outer iteration of the usual MOBIIR algorithm. To improve the performance of the algorithm the outer iteration is reintroduced in which the perturbation equation is recomputed without re-estimating the derivatives and with only updated computed data. The system of quadratic equations is solved using either a modified conjugate gradient descent scheme or a two-step linearized predictor-corrector scheme. A quick method employing the adjoint of the forward operator is used to estimate the derivatives. By solving the nonlinear perturbation equation it is shown that the iterative scheme is able to recover large contrast variations in absorption coefficient with improved noise tolerance in data. This ability has not been possible so far with linear algorithms. This is demonstrated by presenting results of numerical simulations from objects with inhomogeneous inclusions in absorption coefficient with different contrasts and shapes.
Iterative Method for Solving the Second Boundary Value Problem for Biharmonic-Type Equation
Directory of Open Access Journals (Sweden)
Dang Quang A.
2012-01-01
Full Text Available Solving boundary value problems (BVPs for the fourth-order differential equations by the reduction of them to BVPs for the second-order equations with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by ourselves in recent works, we construct iterative method for the second BVP for biharmonic-type equation, which describes the deflection of a plate resting on a biparametric elastic foundation. The convergence rate of the method is established. The optimal value of the iterative parameter is found. Several numerical examples confirm the efficiency of the proposed method.
Imaginary Time Step Method to Solve the Dirac Equation with Nonlocal Potential
International Nuclear Information System (INIS)
Zhang Ying; Liang Haozhao; Meng Jie
2009-01-01
The imaginary time step (ITS) method is applied to solve the Dirac equation with nonlocal potentials in coordinate space. Taking the nucleus 12 C as an example, even with nonlocal potentials, the direct ITS evolution for the Dirac equation still meets the disaster of the Dirac sea. However, following the recipe in our former investigation, the disaster can be avoided by the ITS evolution for the corresponding Schroedinger-like equation without localization, which gives the convergent results exactly the same with those obtained iteratively by the shooting method with localized effective potentials.
International Nuclear Information System (INIS)
Scannapieco, A.J.; Cranfill, C.W.
1978-11-01
There now exists an inertial confinement stability code called DOC, which runs as a postprocessor. DOC (a code that has evolved from a previous code, PANSY) is a spherical harmonic linear stability code that integrates, in time, a set of Lagrangian perturbation equations. Effects due to real equations of state, asymmetric energy deposition, thermal conduction, shock propagation, and a time-dependent zeroth-order state are handled in the code. We present here a detailed derivation of the physical equations that are solved in the code
Bandele, Samuel Oye; Adekunle, Adeyemi Suraju
2015-01-01
The study was conducted to design, develop and test a c++ application program CAP-QUAD for solving quadratic equation in elementary school in Nigeria. The package was developed in c++ using object-oriented programming language, other computer program that were also utilized during the development process is DevC++ compiler, it was used for…
Solving the Helmholtz equation in terms of amplitude and phase; revisited
Wijnant, Y. H.; De Vries, F. H.; Sas, Paul
2014-01-01
In a paper presented at the ISMA conference in 2008 [2], we proposed to solve the Helmholtz equation in terms of the pressure amplitude and phase instead of the complex pressure itself. The major advantage of this approach is the large reduction of the number of degrees of freedom, needed to
Lin, Pao-Ping; Hsieh, Che-Jen
1993-01-01
Describes the Geometer's Sketchpad, a geometric construction kit composed of three manipulatable, dynamic, linked, multiple representation environments: the coordinate system, formulas, and graphs. Examines the use of the environments for studying parameter effects of linear and quadratic functions and for solving linear equations. (MDH)
A novel approach to solve nonlinear Fredholm integral equations of the second kind.
Li, Hu; Huang, Jin
2016-01-01
In this paper, we present a novel approach to solve nonlinear Fredholm integral equations of the second kind. This algorithm is constructed by the integral mean value theorem and Newton iteration. Convergence and error analysis of the numerical solutions are given. Moreover, Numerical examples show the algorithm is very effective and simple.
New approach to solve fully fuzzy system of linear equations using ...
Indian Academy of Sciences (India)
... double parametric form of fuzzy numbers converts the n×n fully fuzzy system of linear equations to a crisp system of same order. Triangular and trapezoidal convex normalized fuzzy sets are used for the present analysis. Known example problems are solved to illustrate the efficacy and reliability of the proposed methods.
Pashazadeh Atabakan, Z.; Kılıçman, A.; Kazemi Nasab, A.
2012-01-01
A modification of homotopy analysis method (HAM) known as spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations. Some examples are given in order to test the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to SHAM results and exact solutions.
Solving Second-Order Ordinary Differential Equations without Using Complex Numbers
Kougias, Ioannis E.
2009-01-01
Ordinary differential equations (ODEs) is a subject with a wide range of applications and the need of introducing it to students often arises in the last year of high school, as well as in the early stages of tertiary education. The usual methods of solving second-order ODEs with constant coefficients, among others, rely upon the use of complex…
Directory of Open Access Journals (Sweden)
Ai-Min Yang
2014-01-01
Full Text Available The local fractional Laplace variational iteration method was applied to solve the linear local fractional partial differential equations. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. The nondifferentiable approximate solutions are obtained and their graphs are also shown.
Magruder, Robin Lee
2012-01-01
The purpose of this embedded quasi-experimental mixed methods research was to use solving simple linear equations as the lens for looking at the effectiveness of concrete and virtual manipulatives as compared to a control group using learning methods without manipulatives. Further, the researcher wanted to investigate unique benefits and drawbacks…
FMCEIR: a Monte Carlo program for solving the stationary neutron and gamma transport equation
International Nuclear Information System (INIS)
Taormina, A.
1978-05-01
FMCEIR is a three-dimensional Monte Carlo program for solving the stationary neutron and gamma transport equation. It is used to study the problem of neutron and gamma streaming in the GCFR and HHT reactor channels. (G.T.H.)
Optimal Homotopy Asymptotic Method for Solving System of Fredholm Integral Equations
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Bahman Ghazanfari
2013-08-01
Full Text Available In this paper, optimal homotopy asymptotic method (OHAM is applied to solve system of Fredholm integral equations. The effectiveness of optimal homotopy asymptotic method is presented. This method provides easy tools to control the convergence region of approximating solution series wherever necessary. The results of OHAM are compared with homotopy perturbation method (HPM and Taylor series expansion method (TSEM.
A theory of solving TAP equations for Ising models with general invariant random matrices
DEFF Research Database (Denmark)
Opper, Manfred; Çakmak, Burak; Winther, Ole
2016-01-01
We consider the problem of solving TAP mean field equations by iteration for Ising models with coupling matrices that are drawn at random from general invariant ensembles. We develop an analysis of iterative algorithms using a dynamical functional approach that in the thermodynamic limit yields...
New numerical methods for solving the time-dependent Maxwell equations
De Raedt, H; Kole, JS; Michielsen, KFL; Figge, MT; Berz, M; Makino, K
2005-01-01
We review some recent developments in numerical algorithms to solve the time-dependent Maxwell equations for systems with spatially varying permittivity and permeabilitly. We show that the Suzuki product-formula approach can be used to construct a family of unconditionally stable algorithms, the
Chin-Joe-Kong, M.J.S.; Mulder, W.A.; van Veldhuizen, M.
1999-01-01
The higher-order finite-element scheme with mass lumping for triangles and tetrahedra is an efficient method for solving the wave equation. A number of lower-order elements have already been found. Here the search for elements of higher order is continued. Elements are constructed in a systematic
A Limited Memory BFGS Method for Solving Large-Scale Symmetric Nonlinear Equations
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Xiangrong Li
2014-01-01
Full Text Available A limited memory BFGS (L-BFGS algorithm is presented for solving large-scale symmetric nonlinear equations, where a line search technique without derivative information is used. The global convergence of the proposed algorithm is established under some suitable conditions. Numerical results show that the given method is competitive to those of the normal BFGS methods.
Parallelizing across time when solving time-dependent partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Worley, P.H.
1991-09-01
The standard numerical algorithms for solving time-dependent partial differential equations (PDEs) are inherently sequential in the time direction. This paper describes algorithms for the time-accurate solution of certain classes of linear hyperbolic and parabolic PDEs that can be parallelized in both time and space and have serial complexities that are proportional to the serial complexities of the best known algorithms. The algorithms for parabolic PDEs are variants of the waveform relaxation multigrid method (WFMG) of Lubich and Ostermann where the scalar ordinary differential equations (ODEs) that make up the kernel of WFMG are solved using a cyclic reduction type algorithm. The algorithms for hyperbolic PDEs use the cyclic reduction algorithm to solve ODEs along characteristics. 43 refs.
The use of Galerkin finite-element methods to solve mass-transport equations
Grove, David B.
1977-01-01
The partial differential equation that describes the transport and reaction of chemical solutes in porous media was solved using the Galerkin finite-element technique. These finite elements were superimposed over finite-difference cells used to solve the flow equation. Both convection and flow due to hydraulic dispersion were considered. Linear and Hermite cubic approximations (basis functions) provided satisfactory results: however, the linear functions were computationally more efficient for two-dimensional problems. Successive over relaxation (SOR) and iteration techniques using Tchebyschef polynomials were used to solve the sparce matrices generated using the linear and Hermite cubic functions, respectively. Comparisons of the finite-element methods to the finite-difference methods, and to analytical results, indicated that a high degree of accuracy may be obtained using the method outlined. The technique was applied to a field problem involving an aquifer contaminated with chloride, tritium, and strontium-90. (Woodard-USGS)
Directory of Open Access Journals (Sweden)
A. H. Bhrawy
2014-01-01
Full Text Available One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.
An efficient numerical technique for solving navier-stokes equations for rotating flows
International Nuclear Information System (INIS)
Haroon, T.; Shah, T.M.
2000-01-01
This paper simulates an industrial problem by solving compressible Navier-Stokes equations. The time-consuming tri-angularization process of a large-banded matrix, performed by memory economical Frontal Technique. This scheme successfully reduces the time for I/O operations even for as large as (40, 000 x 40, 000) matrix. Previously, this industrial problem can solved by using modified Newton's method with Gaussian elimination technique for the large matrix. In the present paper, the proposed Frontal Technique is successfully used, together with Newton's method, to solve compressible Navier-Stokes equations for rotating cylinders. By using the Frontal Technique, the method gives the solution within reasonably acceptance computational time. Results are compared with the earlier works done, and found computationally very efficient. Some features of the solution are reported here for the rotating machines. (author)
Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations
Mamat, M.; Dauda, M. K.; Mohamed, M. A. bin; Waziri, M. Y.; Mohamad, F. S.; Abdullah, H.
2018-03-01
Research from the work of engineers, economist, modelling, industry, computing, and scientist are mostly nonlinear equations in nature. Numerical solution to such systems is widely applied in those areas of mathematics. Over the years, there has been significant theoretical study to develop methods for solving such systems, despite these efforts, unfortunately the methods developed do have deficiency. In a contribution to solve systems of the form F(x) = 0, x ∈ Rn , a derivative free method via the classical Davidon-Fletcher-Powell (DFP) update is presented. This is achieved by simply approximating the inverse Hessian matrix with {Q}k+1-1 to θkI. The modified method satisfied the descent condition and possess local superlinear convergence properties. Interestingly, without computing any derivative, the proposed method never fail to converge throughout the numerical experiments. The output is based on number of iterations and CPU time, different initial starting points were used on a solve 40 benchmark test problems. With the aid of the squared norm merit function and derivative-free line search technique, the approach yield a method of solving symmetric systems of nonlinear equations that is capable of significantly reducing the CPU time and number of iteration, as compared to its counterparts. A comparison between the proposed method and classical DFP update were made and found that the proposed methodis the top performer and outperformed the existing method in almost all the cases. In terms of number of iterations, out of the 40 problems solved, the proposed method solved 38 successfully, (95%) while classical DFP solved 2 problems (i.e. 05%). In terms of CPU time, the proposed method solved 29 out of the 40 problems given, (i.e.72.5%) successfully whereas classical DFP solves 11 (27.5%). The method is valid in terms of derivation, reliable in terms of number of iterations and accurate in terms of CPU time. Thus, suitable and achived the objective.
Zúñiga-Aguilar, C. J.; Coronel-Escamilla, A.; Gómez-Aguilar, J. F.; Alvarado-Martínez, V. M.; Romero-Ugalde, H. M.
2018-02-01
In this paper, we approximate the solution of fractional differential equations with delay using a new approach based on artificial neural networks. We consider fractional differential equations of variable order with the Mittag-Leffler kernel in the Liouville-Caputo sense. With this new neural network approach, an approximate solution of the fractional delay differential equation is obtained. Synaptic weights are optimized using the Levenberg-Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional delay differential equations, linear systems with delay, nonlinear systems with delay and a system of differential equations, for instance, the Newton-Leipnik oscillator. The solution of the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method. To show the effectiveness of the proposed neural network, different performance indices were calculated.
Splitting Method for Solving the Coarse-Mesh Discretized Low-Order Quasi-Diffusion Equations
International Nuclear Information System (INIS)
Hiruta, Hikaru; Anistratov, Dmitriy Y.; Adams, Marvin L.
2005-01-01
In this paper, the development is presented of a splitting method that can efficiently solve coarse-mesh discretized low-order quasi-diffusion (LOQD) equations. The LOQD problem can reproduce exactly the transport scalar flux and current. To solve the LOQD equations efficiently, a splitting method is proposed. The presented method splits the LOQD problem into two parts: (a) the D problem that captures a significant part of the transport solution in the central parts of assemblies and can be reduced to a diffusion-type equation and (b) the Q problem that accounts for the complicated behavior of the transport solution near assembly boundaries. Independent coarse-mesh discretizations are applied: the D problem equations are approximated by means of a finite element method, whereas the Q problem equations are discretized using a finite volume method. Numerical results demonstrate the efficiency of the methodology presented. This methodology can be used to modify existing diffusion codes for full-core calculations (which already solve a version of the D problem) to account for transport effects
Solving the Coupled System Improves Computational Efficiency of the Bidomain Equations
Southern, J.A.
2009-10-01
The bidomain equations are frequently used to model the propagation of cardiac action potentials across cardiac tissue. At the whole organ level, the size of the computational mesh required makes their solution a significant computational challenge. As the accuracy of the numerical solution cannot be compromised, efficiency of the solution technique is important to ensure that the results of the simulation can be obtained in a reasonable time while still encapsulating the complexities of the system. In an attempt to increase efficiency of the solver, the bidomain equations are often decoupled into one parabolic equation that is computationally very cheap to solve and an elliptic equation that is much more expensive to solve. In this study, the performance of this uncoupled solution method is compared with an alternative strategy in which the bidomain equations are solved as a coupled system. This seems counterintuitive as the alternative method requires the solution of a much larger linear system at each time step. However, in tests on two 3-D rabbit ventricle benchmarks, it is shown that the coupled method is up to 80% faster than the conventional uncoupled method-and that parallel performance is better for the larger coupled problem.
A parallel algorithm for solving the integral form of the discrete ordinates equations
International Nuclear Information System (INIS)
Zerr, R. J.; Azmy, Y. Y.
2009-01-01
The integral form of the discrete ordinates equations involves a system of equations that has a large, dense coefficient matrix. The serial construction methodology is presented and properties that affect the execution times to construct and solve the system are evaluated. Two approaches for massively parallel implementation of the solution algorithm are proposed and the current results of one of these are presented. The system of equations May be solved using two parallel solvers-block Jacobi and conjugate gradient. Results indicate that both methods can reduce overall wall-clock time for execution. The conjugate gradient solver exhibits better performance to compete with the traditional source iteration technique in terms of execution time and scalability. The parallel conjugate gradient method is synchronous, hence it does not increase the number of iterations for convergence compared to serial execution, and the efficiency of the algorithm demonstrates an apparent asymptotic decline. (authors)
Directory of Open Access Journals (Sweden)
Yongquan Zhou
2013-01-01
Full Text Available In view of the traditional numerical method to solve the nonlinear equations exist is sensitive to initial value and the higher accuracy of defects. This paper presents an invasive weed optimization (IWO algorithm which has population diversity with the heuristic global search of differential evolution (DE algorithm. In the iterative process, the global exploration ability of invasive weed optimization algorithm provides effective search area for differential evolution; at the same time, the heuristic search ability of differential evolution algorithm provides a reliable guide for invasive weed optimization. Based on the test of several typical nonlinear equations and a circle packing problem, the results show that the differential evolution invasive weed optimization (DEIWO algorithm has a higher accuracy and speed of convergence, which is an efficient and feasible algorithm for solving nonlinear systems of equations.
International Nuclear Information System (INIS)
Hernandez-Walls, R; Martín-Atienza, B; Salinas-Matus, M; Castillo, J
2017-01-01
When solving the linear inviscid shallow water equations with variable depth in one dimension using finite differences, a tridiagonal system of equations must be solved. Here we present an approach, which is more efficient than the commonly used numerical method, to solve this tridiagonal system of equations using a recursion formula. We illustrate this approach with an example in which we solve for a rectangular channel to find the resonance modes. Our numerical solution agrees very well with the analytical solution. This new method is easy to use and understand by undergraduate students, so it can be implemented in undergraduate courses such as Numerical Methods, Lineal Algebra or Differential Equations. (paper)
Solving ordinary differential equations by electrical analogy: a multidisciplinary teaching tool
Sanchez Perez, J. F.; Conesa, M.; Alhama, I.
2016-11-01
Ordinary differential equations are the mathematical formulation for a great variety of problems in science and engineering, and frequently, two different problems are equivalent from a mathematical point of view when they are formulated by the same equations. Students acquire the knowledge of how to solve these equations (at least some types of them) using protocols and strict algorithms of mathematical calculation without thinking about the meaning of the equation. The aim of this work is that students learn to design network models or circuits in this way; with simple knowledge of them, students can establish the association of electric circuits and differential equations and their equivalences, from a formal point of view, that allows them to associate knowledge of two disciplines and promote the use of this interdisciplinary approach to address complex problems. Therefore, they learn to use a multidisciplinary tool that allows them to solve these kinds of equations, even students of first course of engineering, whatever the order, grade or type of non-linearity. This methodology has been implemented in numerous final degree projects in engineering and science, e.g., chemical engineering, building engineering, industrial engineering, mechanical engineering, architecture, etc. Applications are presented to illustrate the subject of this manuscript.
Recent advances in marching-on-in-time schemes for solving time domain volume integral equations
Sayed, Sadeed Bin
2015-05-16
Transient electromagnetic field interactions on inhomogeneous penetrable scatterers can be analyzed by solving time domain volume integral equations (TDVIEs). TDVIEs are constructed by setting the summation of the incident and scattered field intensities to the total field intensity on the volumetric support of the scatterer. The unknown can be the field intensity or flux/current density. Representing the total field intensity in terms of the unknown using the relevant constitutive relation and the scattered field intensity in terms of the spatiotemporal convolution of the unknown with the Green function yield the final form of the TDVIE. The unknown is expanded in terms of local spatial and temporal basis functions. Inserting this expansion into the TDVIE and testing the resulting equation at discrete times yield a system of equations that is solved by the marching on-in-time (MOT) scheme. At each time step, a smaller system of equations, termed MOT system is solved for the coefficients of the expansion. The right-hand side of this system consists of the tested incident field and discretized spatio-temporal convolution of the unknown samples computed at the previous time steps with the Green function.
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M. Bishehniasar
2017-01-01
Full Text Available The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs. The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE. Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD method and standard finite difference (SFD technique, which are popular in the literature for solving engineering problems.
Directory of Open Access Journals (Sweden)
Syahirbanun Isa
2018-02-01
Full Text Available In this paper, the solution of fuzzy differential equations is approximated numerically using diagonally implicit multistep block method of order four. The multistep block method is well known as an efficient and accurate method for solving ordinary differential equations, hence in this paper the method will be used to solve the fuzzy initial value problems where the initial value is a symmetric triangular fuzzy interval. The triangular fuzzy number is not necessarily symmetric, however by imposing symmetry the definition of a triangular fuzzy number can be simplified. The symmetric triangular fuzzy interval is a triangular fuzzy interval that has same left and right width of membership function from the center. Due to this, the parametric form of symmetric triangular fuzzy number is simple and the performing arithmetic operations become easier. In order to interpret the fuzzy problems, Seikkala’s derivative approach is implemented. Characterization theorem is then used to translate the problems into a system of ordinary differential equations. The convergence of the introduced method is also proved. Numerical examples are given to investigate the performance of the proposed method. It is clearly shown in the results that the proposed method is comparable and reliable in solving fuzzy differential equations.
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Mohammed Yusuf Waziri
2015-01-01
Full Text Available We suggest a conjugate gradient (CG method for solving symmetric systems of nonlinear equations without computing Jacobian and gradient via the special structure of the underlying function. This derivative-free feature of the proposed method gives it advantage to solve relatively large-scale problems (500,000 variables with lower storage requirement compared to some existing methods. Under appropriate conditions, the global convergence of our method is reported. Numerical results on some benchmark test problems show that the proposed method is practically effective.
Directory of Open Access Journals (Sweden)
Shadan Sadigh Behzadi
2011-12-01
Full Text Available In this paper, Adomian decomposition method (ADM and homotopy analysis method (HAM are proposed to solving the fuzzy nonlinear Volterra-Fredholm integral equation of the second kind$(FVFIE-2$. we convert a fuzzy nonlinear Volterra-Fredholm integral equation to a nonlinear system of Volterra-Fredholm integral equation in crisp case. we use ADM , HAM and find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the nonlinear fuzzy Volterra-Fredholm integral equation. Also, the existence and uniqueness of the solution and convergence of the proposed methods are proved. Examples is given and the results reveal that homotopy analysis method is very effective and simple compared with the Adomian decomposition method.
Equations of motion solved by the Cremmer-Scherk configuration on even-dimensional spheres
Energy Technology Data Exchange (ETDEWEB)
Kihara, Hironobu [3-26-3-104, Minami-Senzoku, Ota, Tokyo 145-0063 (Japan)
2013-01-15
Equations of motion of low-energy effective theories of quantum electrodynamics include infinitely many interaction terms, which make them difficult to solve. The self-duality property has facilitated research on the solutions to these equations. In this paper, equations of motion of systems of non-Abelian gauge fields on even-dimensional spheres are considered. It is demonstrated that the Cremmer-Scherk configuration, which satisfies certain generalized self-duality equations, becomes the classical solution for the class of systems that are given by arbitrary functions of class C{sup 1} of 2m+ 1 quantities. For instance, Lagrangians consisting of multi-trace terms are included in this class. This result is likely to generate several new and interesting directions of research, including the classification of actions with respect to the stability condition against the Cremmer-Scherk configuration.
Equations of motion solved by the Cremmer-Scherk configuration on even-dimensional spheres
Kihara, Hironobu
2013-01-01
Equations of motion of low-energy effective theories of quantum electrodynamics include infinitely many interaction terms, which make them difficult to solve. The self-duality property has facilitated research on the solutions to these equations. In this paper, equations of motion of systems of non-Abelian gauge fields on even-dimensional spheres are considered. It is demonstrated that the Cremmer-Scherk configuration, which satisfies certain generalized self-duality equations, becomes the classical solution for the class of systems that are given by arbitrary functions of class C1 of 2m + 1 quantities. For instance, Lagrangians consisting of multi-trace terms are included in this class. This result is likely to generate several new and interesting directions of research, including the classification of actions with respect to the stability condition against the Cremmer-Scherk configuration.
Perceptual support promotes strategy generation: Evidence from equation solving.
Alibali, Martha W; Crooks, Noelle M; McNeil, Nicole M
2017-08-30
Over time, children shift from using less optimal strategies for solving mathematics problems to using better ones. But why do children generate new strategies? We argue that they do so when they begin to encode problems more accurately; therefore, we hypothesized that perceptual support for correct encoding would foster strategy generation. Fourth-grade students solved mathematical equivalence problems (e.g., 3 + 4 + 5 = 3 + __) in a pre-test. They were then randomly assigned to one of three perceptual support conditions or to a Control condition. Participants in all conditions completed three mathematical equivalence problems with feedback about correctness. Participants in the experimental conditions received perceptual support (i.e., highlighting in red ink) for accurately encoding the equal sign, the right side of the equation, or the numbers that could be added to obtain the correct solution. Following this intervention, participants completed a problem-solving post-test. Among participants who solved the problems incorrectly at pre-test, those who received perceptual support for correctly encoding the equal sign were more likely to generate new, correct strategies for solving the problems than were those who received feedback only. Thus, perceptual support for accurate encoding of a key problem feature promoted generation of new, correct strategies. Statement of Contribution What is already known on this subject? With age and experience, children shift to using more effective strategies for solving math problems. Problem encoding also improves with age and experience. What the present study adds? Support for encoding the equal sign led children to generate correct strategies for solving equations. Improvements in problem encoding are one source of new strategies. © 2017 The British Psychological Society.
AL-Jawary, M. A.; AL-Qaissy, H. R.
2015-04-01
In this paper, we implement the new iterative method proposed by Daftardar-Gejji and Jafari namely new iterative method (DJM) to solve the linear and non-linear Volterra integro-differential equations and systems of linear and non-linear Volterra integro-differential equations. The applications of the DJM for solving the resulting equations of the non-linear Volterra integro-differential equations forms of the Lane-Emden equations are presented. The Volterra integro-differential equations forms of the Lane-Emden equation overcome the singular behaviour at the origin x = 0 of the original differential equation. Some examples are solved and different cases of the Lane-Emden equations of first kind are presented. Moreover, the DJM is applied to solve the system of the linear and non-linear Volterra integro-differential forms of the Lane-Emden equations. The results demonstrate that the method has many merits such as being derivative-free, and overcoming the difficulty arising in calculating Adomian polynomials to handle the non-linear terms in Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier in Variational Iteration Method (VIM) and no need to construct a homotopy in Homotopy Perturbation Method (HPM) and solve the corresponding algebraic equations.
MPFA algorithm for solving stokes-brinkman equations on quadrilateral grids
Iliev, Oleg
2014-01-01
This work is concerned with the development of a robust and accurate numerical method for solving the Stokes-Brinkman system of equations, which describes a free fluid flow coupled with a flow in porous media. Quadrilateral boundary fitted grid with a sophisticated finite volume method, namely MPFA O-method, is used to discretize the system of equations. Numerical results for two examples are presented, namely, channel flow and flow in a ring with a rolled porous medium. © Springer International Publishing Switzerland 2014.
An Adaptive Observer-Based Algorithm for Solving Inverse Source Problem for the Wave Equation
Asiri, Sharefa M.
2015-08-31
Observers are well known in control theory. Originally designed to estimate the hidden states of dynamical systems given some measurements, the observers scope has been recently extended to the estimation of some unknowns, for systems governed by partial differential equations. In this paper, observers are used to solve inverse source problem for a one-dimensional wave equation. An adaptive observer is designed to estimate the state and source components for a fully discretized system. The effectiveness of the algorithm is emphasized in noise-free and noisy cases and an insight on the impact of measurements’ size and location is provided.
Directory of Open Access Journals (Sweden)
C. Ünlü
2013-01-01
Full Text Available A modification of the variational iteration method (VIM for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE obtained using the traditional variational iteration method give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. Illustrative examples are presented to show the validity of this modification.
SOME CONSIDERATIONS REGARDING ANALITICAL SOLVING OF THE REYNOLD’S EQUATION FROM FACE SEAL
Directory of Open Access Journals (Sweden)
Nicolae POPA
2010-10-01
Full Text Available Taking into consideration a mechanical face seal which runs in hydrodynamic duty, we intend to understand what’s going on under its surfaces. Until nowadays the experimental study to measure the parameters of film between two seal surfaces is very difficult. To explain these phenomena from this interface is necessary to know the theoretical pressures. The paper uses a primary seal model where the seal surfaces are considered with misalignment but the ring centre distance is constant. Considering some simplified hypothesis it determines the Reynold’s equation, it’s solving leading to the pressure equation of sealing interface.
Khader, M M
2013-10-01
In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.
He's variational iteration method for solving a semi-linear inverse parabolic equation
International Nuclear Information System (INIS)
Varedi, S.M.; Hosseini, M.J.; Rahimi, M.; Ganji, D.D.
2007-01-01
Most scientific problems and physical phenomena occur nonlinearly. Except in a limited number of these problems, we have difficulty in finding their exact analytical solutions. A new analytical method called He's variational iteration method (VIM) is introduced to be applied to solve nonlinear equations. In this work VIM is used for finding the solution of a semi-linear inverse parabolic equation. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers can be identified optimally via the variational theory. The results are compared with the exact solutions
A Compact Numerical Implementation for Solving Stokes Equations Using Matrix-vector Operations
Zhang, Tao
2015-06-01
In this work, a numerical scheme is implemented to solve Stokes equations based on cell-centered finite difference over staggered grid. In this scheme, all the difference operations have been vectorized thereby eliminating loops. This is particularly important when using programming languages that require interpretations, e.g., MATLAB and Python. Using this scheme, the execution time becomes significantly smaller compared with non-vectorized operations and also become comparable with those languages that require no repeated interpretations like FORTRAN, C, etc. This technique has also been applied to Navier-Stokes equations under laminar flow conditions.
Directory of Open Access Journals (Sweden)
Sedigheh Farzaneh Javan
2017-01-01
Full Text Available A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second-kind nonlinear integral equations. In this case, the Gram-Schmidt process is substituted by another process so that a satisfactory result is obtained. In this method, the solution is expressed in the form of a series. Furthermore, the convergence of the proposed technique is proved. In order to illustrate the effectiveness and efficiency of the method, four sample integral equations arising in electromagnetics are solved via the given algorithm.
Application of He's homotopy perturbation method for solving fractional Fokker-Planck equations
Energy Technology Data Exchange (ETDEWEB)
Mousa, Mohamed M. [Dept. of Basic Science, Benha Higher Inst. of Tech., Benha Univ. (Egypt); Dept. of Mechanics, al-Farabi Kazakh National Univ., Almaty (Kazakhstan); Kaltayev, Aidarkhan [Dept. of Mechanics, al-Farabi Kazakh National Univ., Almaty (Kazakhstan)
2009-12-15
The fractional Fokker-Planck equation (FFPE) has been used in many physical transport problems which take place under the influence of an external force field and other important applications in various areas of engineering and physics. In this paper, by means of the homotopy perturbation method (HPM), exact and approximate solutions are obtained for two classes of the FFPE initial value problems. The method gives an analytic solution in the form of a convergent series with easily computed components. The obtained results show that the HPM is easy to implement, accurate and reliable for solving FFPEs. The method introduces a promising tool for solving other types of differential equation with fractional order derivatives. (orig.)
International Nuclear Information System (INIS)
Lee, Goung Jin; Chang, Soon Heung
1988-01-01
In solving partial differential equations using finite element method, the great parts of the computing time is taken to calculate the local element matrices. Also the much programming efforts are taken for the local element matrices calculations. To reduce the computing time and the efforts of programming, local elements matrices are calculated by symbolic manipulation method. In this study, symbolic manipulation code REDUCE 3.2 is used. As a results, Fortran subroutine form of local element matrices package is obtained. Using this package, programming efforts would be much reduced. Also the computing time is greatly reduced using the developed package. As a conclusion, it can be said that the developed method can be used to solve the partial differential equation with the less computing times and the less programming efforts than the conventional method
Chanthrasuwan, Maveeka; Asri, Nur Asreenawaty Mohd; Hamid, Nur Nadiah Abd; Majid, Ahmad Abd.; Azmi, Amirah
2017-08-01
The cubic B-spline and cubic trigonometric B-spline functions are used to set up the collocation in finding solutions for the Buckmaster equation. These splines are applied as interpolating functions in the spatial dimension while the finite difference method (FDM) is used to discretize the time derivative. The Buckmaster equation is linearized using Taylor's expansion and solved using two schemes, namely Crank-Nicolson and fully implicit. The von Neumann stability analysis is carried out on the two schemes and they are shown to be conditionally stable. In order to demonstrate the capability of the schemes, some problems are solved and compared with analytical and FDM solutions. The proposed methods are found to generate more accurate results than the FDM.
Performance of a parallel algorithm for solving the neutron diffusion equation on the hypercube
International Nuclear Information System (INIS)
Kirk, B.L.; Azmy, Y.Y.
1989-01-01
The one-group, steady state neutron diffusion equation in two- dimensional Cartesian geometry is solved using the nodal method technique. By decoupling sets of equations representing the neutron current continuity along the length of rows and columns of computational cells a new iterative algorithm is derived that is more suitable to solving large practical problems. This algorithm is highly parallelizable and is implemented on the Intel iPSC/2 hypercube in three versions which differ essentially in the total size of communicated data. Even though speedup was achieved, the efficiency is very low when many processors are used leading to the conclusion that the hypercube is not as well suited for this algorithm as shared memory machines. 10 refs., 1 fig., 3 tabs
Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations
Directory of Open Access Journals (Sweden)
Abdelkawy M.A.
2016-01-01
Full Text Available We introduce a new numerical algorithm for solving one-dimensional time-fractional Tricomi-type equations (T-FTTEs. We used the shifted Jacobi polynomials as basis functions and the derivatives of fractional is evaluated by the Caputo definition. The shifted Jacobi Gauss-Lobatt algorithm is used for the spatial discretization, while the shifted Jacobi Gauss-Radau algorithmis applied for temporal approximation. Substituting these approximations in the problem leads to a system of algebraic equations that greatly simplifies the problem. The proposed algorithm is successfully extended to solve the two-dimensional T-FTTEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.
Directory of Open Access Journals (Sweden)
E. Fathizadeh
2017-01-01
Full Text Available We use a computational method based on rational Haar wavelet for solving nonlinear fractional integro-differential equations. To this end, we apply the operational matrix of fractional integration for rational Haar wavelet. Also, to show the efficiency of the proposed method, we solve particularly population growth model and Abel integral equations and compare the numerical results with the exact solutions.
Khotimah, Rita Pramujiyanti; Masduki
2016-01-01
Differential equations is a branch of mathematics which is closely related to mathematical modeling that arises in real-world problems. Problem solving ability is an essential component to solve contextual problem of differential equations properly. The purposes of this study are to describe contextual teaching and learning (CTL) model in…
On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation
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Hameed Husam Hameed
2015-01-01
Full Text Available We develop the Newton-Kantorovich method to solve the system of 2×2 nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.
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Xiaolin Zhu
2014-01-01
Full Text Available This paper studies the T-stability of the Heun method and balanced method for solving stochastic differential delay equations (SDDEs. Two T-stable conditions of the Heun method are obtained for two kinds of linear SDDEs. Moreover, two conditions under which the balanced method is T-stable are obtained for two kinds of linear SDDEs. Some numerical examples verify the theoretical results proposed.
Rebenda, Josef; Šmarda, Zdeněk
2017-07-01
In the paper, we propose a correct and efficient semi-analytical approach to solve initial value problem for systems of functional differential equations with delay. The idea is to combine the method of steps and differential transformation method (DTM). In the latter, formulas for proportional arguments and nonlinear terms are used. An example of using this technique for a system with constant and proportional delays is presented.
A Generalized FDM for solving the Poisson's Equation on 3D Irregular Domains
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J. Izadian
2014-01-01
Full Text Available In this paper a new method for solving the Poisson's equation with Dirichlet conditions on irregular domains is presented. For this purpose a generalized finite differences method is applied for numerical differentiation on irregular meshes. Three examples on cylindrical and spherical domains are considered. The numerical results are compared with analytical solution. These results show the performance and efficiency of the proposed method.
Solving the two-dimensional Schrödinger equation using basis ...
Indian Academy of Sciences (India)
Ihab H Naeim
2017-10-19
Oct 19, 2017 ... PACS Nos 02.30.Mv; 02.60.Dc; 03.65.−w; 03.65.Ge. 1. Introduction. A common textbook quantum physics/chemistry approach to solve the Schrödinger equation in those cases where no analytical solution is available is to utilize matrix mechanics [1]. In atomic and molecular electronic structure calculations ...
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Salih Yalcinbas
2016-01-01
Full Text Available In this paper, a new collocation method based on the Fibonacci polynomials is introduced to solve the high-order linear Volterra integro-differential equations under the conditions. Numerical examples are included to demonstrate the applicability and validity of the proposed method and comparisons are made with the existing results. In addition, an error estimation based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation.
Comparison of numerical approaches to solve a Poincare-covariant Faddeev equation
International Nuclear Information System (INIS)
Alkofer, R.; Eichmann, G.; Krassnigg, A.; Schwinzerl, M.
2006-01-01
Full text: The quark core of Baryons can be described with the help of the numerical solution of the Poincare-Faddeev equation. Hereby the used elements, as e.g. the quark propagator are taken from non-perturbative studies of Landau gauge QCD. Different numerical approaches to solve in this way the relativistic three quark problem are compared and benchmarked results for the efficiency of different algorithms are presented. (author)
New version of the RADUGA system for solving transport equations in the R-Z geometry
International Nuclear Information System (INIS)
Bass, L.P.; Germogenova, T.A.; Goncharov, A.N.; Petrulevich, A.A.; Khmylev, A.N.
1987-01-01
The RADUGA-3 version of the RADUGA modular program used to solve the multigroup system of equations of radiation transport with anisotropic scattering by the discrete ordinate method in two-dimensional R-Z-geometry, is described. Modules, introduced into the new version, broaden the program possibilities and allow to improve calculational accuracy and to reduce essentially (by 2-5 times) calculation time
On the equivalence of Gaussian elimination and Gauss-Jordan reduction in solving linear equations
Tsao, Nai-Kuan
1989-01-01
A novel general approach to round-off error analysis using the error complexity concepts is described. This is applied to the analysis of the Gaussian Elimination and Gauss-Jordan scheme for solving linear equations. The results show that the two algorithms are equivalent in terms of our error complexity measures. Thus the inherently parallel Gauss-Jordan scheme can be implemented with confidence if parallel computers are available.
Directory of Open Access Journals (Sweden)
Sawicki Dominik
2015-09-01
Full Text Available One of the most popular applications of high power lasers is heating of the surface layer of a material, in order to change its properties. Numerical methods allow an easy and fast way to simulate the heating process inside of the material. The most popular numerical methods FEM and BEM, used to simulate this kind of processes have one fundamental defect, which is the necessity of discretization of the boundary or the domain. An alternative to avoid the mentioned problem are parametric integral equations systems (PIES, which do not require classical discretization of the boundary and the domain while being numerically solved. PIES method was previously used with success to solve steady-state problems, as well as transient heat transfer problems. The purpose of this paper is to test the efficacy of the PIES method with time discretization in solving problem of laser heating of a material, with different pulse shape approximation functions.
Energy Technology Data Exchange (ETDEWEB)
Gene Golub; Kwok Ko
2009-03-30
The solutions of sparse eigenvalue problems and linear systems constitute one of the key computational kernels in the discretization of partial differential equations for the modeling of linear accelerators. The computational challenges faced by existing techniques for solving those sparse eigenvalue problems and linear systems call for continuing research to improve on the algorithms so that ever increasing problem size as required by the physics application can be tackled. Under the support of this award, the filter algorithm for solving large sparse eigenvalue problems was developed at Stanford to address the computational difficulties in the previous methods with the goal to enable accelerator simulations on then the world largest unclassified supercomputer at NERSC for this class of problems. Specifically, a new method, the Hemitian skew-Hemitian splitting method, was proposed and researched as an improved method for solving linear systems with non-Hermitian positive definite and semidefinite matrices.
High-order accurate methods for solving the time-harmonic Maxwell's equations
Wilcox, Lucas Charles
Maxwell's equations are the partial differential equations describing electromagnetism. They can be used to model electric and magnetic fields in different materials from light in fiber optic cables to radar waves bouncing off a stealth fighter jet. In problems with electromagnetic radiation of a single frequency Maxwell's equations may be reduced to their time-harmonic form. Further simplifying the problem a multilayer boundary variation method for the forward modeling of multilayered diffraction optics is presented. This approach enables fast and high-order accurate modeling of periodic transmission optics consisting of an arbitrary number of materials and interfaces of general shape subject to plane wave illumination or, by solving a sequence of problems, illumination by beams. The key developments of the algorithm are discussed as are details of an efficient implementation. Numerous comparisons with exact solutions and highly accurate direct solutions confirm the accuracy, versatility, and efficiency of the proposed method. The high accuracy of the method is leveraged to solve an application involving the in-coupling process for grating-coupled planar optical waveguide sensors. For more general solutions of the time-harmonic Maxwell's equations an hp-adaptive discontinuous Galerkin finite element method is studied. The discontinuous Galerkin finite element method is a general method for solving partial differential equations that has had success with time evolution problems. The application to time-harmonic problems is a new and developing area of research. As a first step, an overlapping Schwarz method for the discontinuous Galerkin discretization of the indefinite Helmholtz equation is examined. For an hp-adaptive method to be successful an error indicator is required to determine the areas of the computational domain that need increased resolution. The use of adjoint based error indicators is explored through solving the time-harmonic Maxwell's equations for
An efficient numerical method for solving the Boltzmann equation in multidimensions
Dimarco, Giacomo; Loubère, Raphaël; Narski, Jacek; Rey, Thomas
2018-01-01
In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) (Dimarco and Loubère, 2013 [26]) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3 D × 3 D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.
A Multiagent Transfer Function Neuroapproach to Solve Fuzzy Riccati Differential Equations
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Mohammad Shazri Shahrir
2014-01-01
Full Text Available A numerical solution of fuzzy quadratic Riccati differential equation is estimated using a proposed new approach for neural networks (NN. This proposed new approach provides different degrees of polynomial subspaces for each of the transfer function. This multitude of transfer functions creates unique “agents” in the structure of the NN. Hence it is named as multiagent neuroapproach (multiagent NN. Previous works have shown that results using Runge-Kutta 4th order (RK4 are reliable. The results can be achieved by solving the 1st order nonlinear differential equation (ODE that is found commonly in Riccati differential equation. Multiagent NN shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over Mabood et al. (2013, RK-4, and the existing neuromethod (NM. Numerical examples are discussed to illustrate the proposed method.
Orbital HP-Clouds for Solving Schr?dinger Equation inQuantum Mechanics
Energy Technology Data Exchange (ETDEWEB)
Chen, J; Hu, W; Puso, M
2006-10-19
Solving Schroedinger equation in quantum mechanics presents a challenging task in numerical methods due to the high order behavior and high dimension characteristics in the wave functions, in addition to the highly coupled nature between wave functions. This work introduces orbital and polynomial enrichment functions to the partition of unity for solution of Schroedinger equation under the framework of HP-Clouds. An intrinsic enrichment of orbital function and extrinsic enrichment of monomial functions are proposed. Due to the employment of higher order basis functions, a higher order stabilized conforming nodal integration is developed. The proposed methods are implemented using the density functional theory for solution of Schroedinger equation. Analysis of several single and multi-electron/nucleus structures demonstrates the effectiveness of the proposed method.
Numerical method for solving stochastic differential equations with Poissonian white shot noise.
Kim, Changho; Lee, Eok Kyun; Hänggi, Peter; Talkner, Peter
2007-07-01
We propose a numerical integration scheme to solve stochastic differential equations driven by Poissonian white shot noise. Our formula, which is based on an integral equation, which is equivalent to the stochastic differential equation, utilizes a discrete time approximation with fixed integration time step. We show that our integration formula approaches the Euler formula if the Poissonian noise approaches the Gaussian white noise. The accuracy and efficiency of the proposed algorithm are examined by studying the dynamics of an overdamped particle driven by Poissonian white shot noise in a spatially periodic potential. We find that the accuracy of the proposed algorithm only weakly depends on the parameters characterizing the Poissonian white shot noise; this holds true even if the limit of Gaussian white noise is approached.
Solving the Vlasov equation in two spatial dimensions with the Schrödinger method
Kopp, Michael; Vattis, Kyriakos; Skordis, Constantinos
2017-12-01
We demonstrate that the Vlasov equation describing collisionless self-gravitating matter may be solved with the so-called Schrödinger method (ScM). With the ScM, one solves the Schrödinger-Poisson system of equations for a complex wave function in d dimensions, rather than the Vlasov equation for a 2 d -dimensional phase space density. The ScM also allows calculating the d -dimensional cumulants directly through quasilocal manipulations of the wave function, avoiding the complexity of 2 d -dimensional phase space. We perform for the first time a quantitative comparison of the ScM and a conventional Vlasov solver in d =2 dimensions. Our numerical tests were carried out using two types of cold cosmological initial conditions: the classic collapse of a sine wave and those of a Gaussian random field as commonly used in cosmological cold dark matter N-body simulations. We compare the first three cumulants, that is, the density, velocity and velocity dispersion, to those obtained by solving the Vlasov equation using the publicly available code ColDICE. We find excellent qualitative and quantitative agreement between these codes, demonstrating the feasibility and advantages of the ScM as an alternative to N-body simulations. We discuss, the emergence of effective vorticity in the ScM through the winding number around the points where the wave function vanishes. As an application we evaluate the background pressure induced by the non-linearity of large scale structure formation, thereby estimating the magnitude of cosmological backreaction. We find that it is negligibly small and has time dependence and magnitude compatible with expectations from the effective field theory of large scale structure.
A toolbox to solve coupled systems of differential and difference equations
International Nuclear Information System (INIS)
Ablinger, Jakob; Schneider, Carsten; Bluemlein, Johannes; Freitas, Abilio de
2016-01-01
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do not request special choices of bases. Here we assume that the desired solution has a power series representation and we seek for the coefficients in closed form. In particular, if the coefficients depend on a small parameter ε (the dimensional parameter), we assume that the coefficients themselves can be expanded in formal Laurent series w.r.t. ε and we try to compute the first terms in closed form. More precisely, we have a decision algorithm which solves the following problem: if the terms can be represented by an indefinite nested hypergeometric sum expression (covering as special cases the harmonic sums, cyclotomic sums, generalized harmonic sums or nested binomial sums), then we can calculate them. If the algorithm fails, we obtain a proof that the terms cannot be represented by the class of indefinite nested hypergeometric sum expressions. Internally, this problem is reduced by holonomic closure properties to solving a coupled system of linear difference equations. The underlying method in this setting relies on decoupling algorithms, difference ring algorithms and recurrence solving. We demonstrate by a concrete example how this algorithm can be applied with the new Mathematica package SolveCoupledSystem which is based on the packages Sigma, HarmonicSums and OreSys. In all applications the representation in x-space is obtained as an iterated integral representation over general alphabets, generalizing Poincare iterated integrals.
Different seeds to solve the equations of stochastic point kinetics using the Euler-Maruyama method
International Nuclear Information System (INIS)
Suescun D, D.; Oviedo T, M.
2017-09-01
In this paper, a numerical study of stochastic differential equations that describe the kinetics in a nuclear reactor is presented. These equations, known as the stochastic equations of punctual kinetics they model temporal variations in neutron population density and concentrations of deferred neutron precursors. Because these equations are probabilistic in nature (since random oscillations in the neutrons and population of precursors were considered to be approximately normally distributed, and these equations also possess strong coupling and stiffness properties) the proposed method for the numerical simulations is the Euler-Maruyama scheme that provides very good approximations for calculating the neutron population and concentrations of deferred neutron precursors. The method proposed for this work was computationally tested for different seeds, initial conditions, experimental data and forms of reactivity for a group of precursors and then for six groups of deferred neutron precursors at each time step with 5000 Brownian movements per seed. In a paper reported in the literature, the Euler-Maruyama method was proposed, but there are many doubts about the reported values, in addition to not reporting the seed used, so in this work is expected to rectify the reported values. After taking the average of the different seeds used to generate the pseudo-random numbers the results provided by the Euler-Maruyama scheme will be compared in mean and standard deviation with other methods reported in the literature and results of the deterministic model of the equations of the punctual kinetics. This comparison confirms in particular that the Euler-Maruyama scheme is an efficient method to solve the equations of stochastic point kinetics but different from the values found and reported by another author. The Euler-Maruyama method is simple and easy to implement, provides acceptable results for neutron population density and concentration of deferred neutron precursors and
Generalized finite-difference time-domain schemes for solving nonlinear Schrodinger equations
Moxley, Frederick Ira, III
The nonlinear Schrodinger equation (NLSE) is one of the most widely applicable equations in physical science, and characterizes nonlinear dispersive waves, optics, water waves, and the dynamics of molecules. The NLSE satisfies many mathematical conservation laws. Moreover, due to the nonlinearity, the NLSE often requires a numerical solution, which also satisfies the conservation laws. Some of the more popular numerical methods for solving the NLSE include the finite difference, finite element, and spectral methods such as the pseudospectral, split-step with Fourier transform, and integrating factor coupled with a Fourier transform. With regard to the finite difference and finite element methods, higher-order accurate and stable schemes are often required to solve a large-scale linear system. Conversely, spectral methods via Fourier transforms for space discretization coupled with Runge-Kutta methods for time stepping become too complex when applied to multidimensional problems. One of the most prevalent challenges in developing these numerical schemes is that they satisfy the conservation laws. The objective of this dissertation was to develop a higher-order accurate and simple finite difference scheme for solving the NLSE. First, the wave function was split into real and imaginary components and then substituted into the NLSE to obtain coupled equations. These components were then approximated using higher-order Taylor series expansions in time, where the derivatives in time were replaced by the derivatives in space via the coupled equations. Finally, the derivatives in space were approximated using higher-order accurate finite difference approximations. As such, an explicit and higher order accurate finite difference scheme for solving the NLSE was obtained. This scheme is called the explicit generalized finite-difference time-domain (explicit G-FDTD). For purposes of completeness, an implicit G-FDTD scheme for solving the NLSE was also developed. In this
Performance prediction of gas turbines by solving a system of non-linear equations
Energy Technology Data Exchange (ETDEWEB)
Kaikko, J.
1998-09-01
This study presents a novel method for implementing the performance prediction of gas turbines from the component models. It is based on solving the non-linear set of equations that corresponds to the process equations, and the mass and energy balances for the engine. General models have been presented for determining the steady state operation of single components. Single and multiple shad arrangements have been examined with consideration also being given to heat regeneration and intercooling. Emphasis has been placed upon axial gas turbines of an industrial scale. Applying the models requires no information of the structural dimensions of the gas turbines. On comparison with the commonly applied component matching procedures, this method incorporates several advantages. The application of the models for providing results is facilitated as less attention needs to be paid to calculation sequences and routines. Solving the set of equations is based on zeroing co-ordinate functions that are directly derived from the modelling equations. Therefore, controlling the accuracy of the results is easy. This method gives more freedom for the selection of the modelling parameters since, unlike for the matching procedures, exchanging these criteria does not itself affect the algorithms. Implicit relationships between the variables are of no significance, thus increasing the freedom for the modelling equations as well. The mathematical models developed in this thesis will provide facilities to optimise the operation of any major gas turbine configuration with respect to the desired process parameters. The computational methods used in this study may also be adapted to any other modelling problems arising in industry. (orig.) 36 refs.
Solving the Einstein constraint equations on multi-block triangulations using finite element methods
International Nuclear Information System (INIS)
Korobkin, Oleg; Pazos, Enrique; Aksoylu, Burak; Holst, Michael; Tiglio, Manuel
2009-01-01
In order to generate initial data for nonlinear relativistic simulations, one needs to solve the Einstein constraints, which can be cast into a coupled set of nonlinear elliptic equations. Here we present an approach for solving these equations on three-dimensional multi-block domains using finite element methods. We illustrate our approach on a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor ψ. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, after constructing the initial data we evolve them in time using a high-order finite-differencing multi-block approach and extract the gravitational waves from the numerical solution.
New classes of potentials for which the radial Schroedinger equation can be solved at zero energy
International Nuclear Information System (INIS)
Chadan, Khosrow; Kobayashi, Reido
2006-01-01
Given two spherically symmetric and short-range potentials V 0 and V 1 for which the radial Schroedinger equation can be solved explicitly at zero energy, we show how to construct a new potential V for which the radial equation can again be solved explicitly at zero energy. The new potential and its corresponding wavefunction are given explicitly in terms of V 0 and V 1 , and their corresponding wavefunctions ψ 0 and ψ 1 . V 0 must be such that it sustains no bound states (either repulsive, or attractive but weak). However, V 1 can sustain any (finite) number of bound states. The new potential V has the same number of bound states, by construction, but the corresponding (negative) energies are, of course, different. Once this is achieved, one can start then from V 0 and V, and construct a new potential V-bar for which the radial equation is again solvable explicitly. And the process can be repeated indefinitely. We exhibit first the construction, and the proof of its validity, for regular short-range potentials, i.e. those for which rV 0 (r) and rV 1 (r) are L 1 at the origin. It is then seen that the construction extends automatically to potentials which are singular at r = 0. It can also be extended to V 0 long range (Coulomb, etc). We finally give several explicit examples
Solving the Einstein constraint equations on multi-block triangulations using finite element methods
Energy Technology Data Exchange (ETDEWEB)
Korobkin, Oleg; Pazos, Enrique [Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803 (United States); Aksoylu, Burak [Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803 (United States); Holst, Michael [Department of Mathematics, University of California at San Diego 9500 Gilman Drive La Jolla, CA 92093-0112 (United States); Tiglio, Manuel [Department of Physics, University of Maryland, College Park, MD 20742 (United States)
2009-07-21
In order to generate initial data for nonlinear relativistic simulations, one needs to solve the Einstein constraints, which can be cast into a coupled set of nonlinear elliptic equations. Here we present an approach for solving these equations on three-dimensional multi-block domains using finite element methods. We illustrate our approach on a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor psi. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, after constructing the initial data we evolve them in time using a high-order finite-differencing multi-block approach and extract the gravitational waves from the numerical solution.
International Nuclear Information System (INIS)
Kirk, B.L.; Azmy, Y.Y.
1992-01-01
In this paper the one-group, steady-state neutron diffusion equation in two-dimensional Cartesian geometry is solved using the nodal integral method. The discrete variable equations comprise loosely coupled sets of equations representing the nodal balance of neutrons, as well as neutron current continuity along rows or columns of computational cells. An iterative algorithm that is more suitable for solving large problems concurrently is derived based on the decomposition of the spatial domain and is accelerated using successive overrelaxation. This algorithm is very well suited for parallel computers, especially since the spatial domain decomposition occurs naturally, so that the number of iterations required for convergence does not depend on the number of processors participating in the calculation. Implementation of the authors' algorithm on the Intel iPSC/2 hypercube and Sequent Balance 8000 parallel computer is presented, and measured speedup and efficiency for test problems are reported. The results suggest that the efficiency of the hypercube quickly deteriorates when many processors are used, while the Sequent Balance retains very high efficiency for a comparable number of participating processors. This leads to the conjecture that message-passing parallel computers are not as well suited for this algorithm as shared-memory machines
Fifth Order Predictor-Corrector Method for Solving Quadratic Riccati Differential Equations
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Gashu Gadisa Kiltu
2017-12-01
Full Text Available In this paper, fifth order predictor-corrector method is presented for solving quadratic Riccati differential equations. First, the interval is discretized and then the method is formulated by using the Newton’s backward difference interpolation formula. The stability and convergence of the method have been investigated. To validate the applicability of the proposed method, three model examples with exact solutions have been considered and numerically solved by using MATLAB software. The numerical results are presented in tables and figures for different values of mesh size h. Pointwise absolute errors and maximum absolute errors are also estimated. Concisely, the present method gives better result than some existing numerical methods reported in the literature.
Schrödinger equation solved for the hydrogen molecule with unprecedented accuracy
Energy Technology Data Exchange (ETDEWEB)
Pachucki, Krzysztof, E-mail: krp@fuw.edu.pl [Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw (Poland); Komasa, Jacek, E-mail: komasa@man.poznan.pl [Faculty of Chemistry, Adam Mickiewicz University, Umultowska 89b, 61-614 Poznań (Poland)
2016-04-28
The hydrogen molecule can be used for determination of physical constants, including the proton charge radius, and for improved tests of the hypothetical long range force between hadrons, which require a sufficiently accurate knowledge of the molecular levels. In this work, we perform the first step toward a significant improvement in theoretical predictions of H{sub 2} and solve the nonrelativistic Schrödinger equation to the unprecedented accuracy of 10{sup −12}. We hope that it will inspire a parallel progress in the spectroscopy of the molecular hydrogen.
A two-qubit photonic quantum processor and its application to solving systems of linear equations
Barz, Stefanie; Kassal, Ivan; Ringbauer, Martin; Lipp, Yannick Ole; Dakić, Borivoje; Aspuru-Guzik, Alán; Walther, Philip
2014-01-01
Large-scale quantum computers will require the ability to apply long sequences of entangling gates to many qubits. In a photonic architecture, where single-qubit gates can be performed easily and precisely, the application of consecutive two-qubit entangling gates has been a significant obstacle. Here, we demonstrate a two-qubit photonic quantum processor that implements two consecutive CNOT gates on the same pair of polarisation-encoded qubits. To demonstrate the flexibility of our system, we implement various instances of the quantum algorithm for solving of systems of linear equations. PMID:25135432
A theory of solving TAP equations for Ising models with general invariant random matrices
DEFF Research Database (Denmark)
Opper, Manfred; Çakmak, Burak; Winther, Ole
2016-01-01
We consider the problem of solving TAP mean field equations by iteration for Ising models with coupling matrices that are drawn at random from general invariant ensembles. We develop an analysis of iterative algorithms using a dynamical functional approach that in the thermodynamic limit yields...... an effective dynamics of a single variable trajectory. Our main novel contribution is the expression for the implicit memory term of the dynamics for general invariant ensembles. By subtracting these terms, that depend on magnetizations at previous time steps, the implicit memory terms cancel making...
Modeling digital pulse waveforms by solving one-dimensional Navier-stokes equations.
Fedotov, Aleksandr A; Akulova, Anna S; Akulov, Sergey A
2016-08-01
Mathematical modeling for composition distal arterial pulse wave in the blood vessels of the upper limbs was considered. Formation of distal arterial pulse wave is represented as a composition of forward and reflected pulse waves propagating along the arterial vessels. The formal analogy between pulse waves propagation along the human arterial system and the propagation of electrical oscillations in electrical transmission lines with distributed parameters was proposed. Dependencies of pulse wave propagation along the human arterial system were obtained by solving the one-dimensional Navier-Stokes equations for a few special cases.
Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations
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Huiping Cao
2014-01-01
Full Text Available Schubert’s method is an extension of Broyden’s method for solving sparse nonlinear equations, which can preserve the zero-nonzero structure defined by the sparse Jacobian matrix and can retain many good properties of Broyden’s method. In particular, Schubert’s method has been proved to be locally and q-superlinearly convergent. In this paper, we globalize Schubert’s method by using a nonmonotone line search. Under appropriate conditions, we show that the proposed algorithm converges globally and superlinearly. Some preliminary numerical experiments are presented, which demonstrate that our algorithm is effective for large-scale problems.
Numerical methods to solve the two-dimensional heat conduction equation
International Nuclear Information System (INIS)
Santos, R.S. dos.
1981-09-01
A class of numerical methods, called 'Hopscotch Algorithms', was used to solve the heat conduction equation in cylindrical geometry. Using a time dependent heat source, the temperature versus time behaviour of cylindric rod was analysed. Numerical simulation was used to study the stability and the convergence of each different method. Another test had the temperature specified on the outer surface as boundary condition. The various Hopscotch methods analysed exhibit differing degrees of accuracy, few of them being so accurate as the ADE method, but requiring more computational operations than the later, were observed. Finally, compared with the so called ODD-EVEN method, two other Hopscotch methods, are more time consuming. (Author) [pt
Sesiano, Jacques
2009-01-01
This text should not be viewed as a comprehensive history of algebra before 1600, but as a basic introduction to the types of problems that illustrate the earliest forms of algebra. It would be particularly useful for an instructor who is looking for examples to help enliven a course on elementary algebra with problems drawn from actual historical texts. -Warren Van Egmond about the French edition for MathSciNet This book does not aim to give an exhaustive survey of the history of algebra up to early modern times but merely to present some significant steps in solving equations and, wherever
Solving Nondifferentiable Nonlinear Equations by New Steffensen-Type Iterative Methods with Memory
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J. P. Jaiswal
2014-01-01
Full Text Available It is attempted to present two derivative-free Steffensen-type methods with memory for solving nonlinear equations. By making use of a suitable self-accelerator parameter in the existing optimal fourth- and eighth-order without memory methods, the order of convergence has been increased without any extra function evaluation. Therefore, its efficiency index is also increased, which is the main contribution of this paper. The self-accelerator parameters are estimated using Newton’s interpolation. To show applicability of the proposed methods, some numerical illustrations are presented.
Directory of Open Access Journals (Sweden)
F. P. Santos
2013-09-01
Full Text Available Direct-quadrature generalized moment based methods were analysed in terms of accuracy, computational cost and robustness for the solution of the population balance problems in the [0,∞ and [0,1] domains. The minimum condition number of the coefficient matrix of their linear system of equations was obtained by global optimization. An heuristic scaling rule from the literature was also evaluated. The results indicate that the methods based on Legendre generalized moments are the most robust for the finite domain problems, while the DQMoM formulation that solves for the abscissas and weights using the heuristic scaling rule is the best for the infinite domain problems.
Newton's method for solving a quadratic matrix equation with special coefficient matrices
International Nuclear Information System (INIS)
Seo, Sang-Hyup; Seo, Jong Hyun; Kim, Hyun-Min
2014-01-01
We consider the iterative method for solving a quadratic matrix equation with special coefficient matrices which arises in the quasi-birth-death problem. In this paper, we show that the elementwise minimal positive solvents to quadratic matrix equations can be obtained using Newton's method. We also prove that the convergence rate of the Newton iteration is quadratic if the Fréchet derivative at the elementwise minimal positive solvent is nonsingular. However, if the Fréchet derivative is singular, the convergence rate is at least linear. Numerical experiments of the convergence rate are given.(This is summarized a paper which is to appear in Honam Mathematical Journal.)
Prediction of Shock Wave Structure in Weakly Ionized Gas Flow by Solving MGD Equation
Deng, Z. T.; Oviedo-Rojas, Ruben; Chow, Alan; Litchford, Ron J.; Cook, Stephen (Technical Monitor)
2002-01-01
This paper reports the recent research results of shockwave structure predictions using a new developed code. The modified Rankine-Hugoniot relations across a standing normal shock wave are discussed and adopted to obtain jump conditions. Coupling a electrostatic body force to the Burnett equations, the weakly ionized flow field across the shock wave was solved. Results indicated that the Modified Rankine-Hugoniot equations for shock wave are valid for a wide range of ionization fraction. However, this model breaks down with small free stream Mach number and with large ionization fraction. The jump conditions also depend on the value of free stream pressure, temperature and density. The computed shock wave structure with ionization provides results, which indicated that shock wave strength may be reduced by existence of weakly ionized gas.
Directory of Open Access Journals (Sweden)
F. F. Ngwane
2015-01-01
Full Text Available We propose a block hybrid trigonometrically fitted (BHT method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs, including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs, such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.
Recent symbolic summation methods to solve coupled systems of differential and difference equations
International Nuclear Information System (INIS)
Schneider, Carsten; Bluemlein, Johannes; Freitas, Abilio de
2014-07-01
We outline a new algorithm to solve coupled systems of differential equations in one continuous variable x (resp. coupled difference equations in one discrete variable N) depending on a small parameter ε: given such a system and given sufficiently many initial values, we can determine the first coefficients of the Laurent-series solutions in ε if they are expressible in terms of indefinite nested sums and products. This systematic approach is based on symbolic summation algorithms in the context of difference rings/fields and uncoupling algorithms. The proposed method gives rise to new interesting applications in connection with integration by parts (IBP) methods. As an illustrative example, we will demonstrate how one can calculate the ε-expansion of a ladder graph with 6 massive fermion lines.
Directory of Open Access Journals (Sweden)
Luis Gavete
2018-01-01
Full Text Available We apply a 3D adaptive refinement procedure using meshless generalized finite difference method for solving elliptic partial differential equations. This adaptive refinement, based on an octree structure, allows adding nodes in a regular way in order to obtain smooth transitions with different nodal densities in the model. For this purpose, we define an error indicator as stop condition of the refinement, a criterion for choosing nodes with the highest errors, and a limit for the number of nodes to be added in each adaptive stage. This kind of equations often appears in engineering problems such as simulation of heat conduction, electrical potential, seepage through porous media, or irrotational flow of fluids. The numerical results show the high accuracy obtained.
Solving differential equations for Feynman integrals by expansions near singular points
Lee, Roman N.; Smirnov, Alexander V.; Smirnov, Vladimir A.
2018-03-01
We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales, i.e. non-trivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer code constructed with the help of our algorithm for a simple example of four-loop generalized sunset integrals with three equal non-zero masses and two zero masses. Our code gives values of the master integrals at any given point on the real axis with a required accuracy and a given order of expansion in the regularization parameter ɛ.
Solving differential equations with unknown constitutive relations as recurrent neural networks
Energy Technology Data Exchange (ETDEWEB)
Hagge, Tobias J.; Stinis, Panagiotis; Yeung, Enoch H.; Tartakovsky, Alexandre M.
2017-12-08
We solve a system of ordinary differential equations with an unknown functional form of a sink (reaction rate) term. We assume that the measurements (time series) of state variables are partially available, and use a recurrent neural network to “learn” the reaction rate from this data. This is achieved by including discretized ordinary differential equations as part of a recurrent neural network training problem. We extend TensorFlow’s recurrent neural network architecture to create a simple but scalable and effective solver for the unknown functions, and apply it to a fedbatch bioreactor simulation problem. Use of techniques from recent deep learning literature enables training of functions with behavior manifesting over thousands of time steps. Our networks are structurally similar to recurrent neural networks, but differ in purpose, and require modified training strategies.
An optimal iterative algorithm to solve Cauchy problem for Laplace equation
Majeed, Muhammad Usman
2015-05-25
An optimal mean square error minimizer algorithm is developed to solve severely ill-posed Cauchy problem for Laplace equation on an annulus domain. The mathematical problem is presented as a first order state space-like system and an optimal iterative algorithm is developed that minimizes the mean square error in states. Finite difference discretization schemes are used to discretize first order system. After numerical discretization algorithm equations are derived taking inspiration from Kalman filter however using one of the space variables as a time-like variable. Given Dirichlet and Neumann boundary conditions are used on the Cauchy data boundary and fictitious points are introduced on the unknown solution boundary. The algorithm is run for a number of iterations using the solution of previous iteration as a guess for the next one. The method developed happens to be highly robust to noise in Cauchy data and numerically efficient results are illustrated.
Validation of a tetrahedral spectral element code for solving the Navier Stokes equation
International Nuclear Information System (INIS)
Niewiadomski, C.; Paraschivoiu, M.
2004-01-01
The tetrahedral spectral element method is considered to solve the incompressible Navier-Stokes equations because it is capable to capture complex geometries and obtain highly accurate solutions. This method allows accuracy improvements both by decreasing the spatial discretization as well as increasing the expansion order. The method is presented here-in as a modification of an standard finite element code. Some recent improvement to the baseline spectral element method for the tetrahedron described in References 3 and 2 are presented. These improvements include: the continuity enforcement procedure avoiding the need to change the global assembly operation and the removal of the reference coordinate system from the elemental evaluations thus simplifying greatly the method. A study is performed on the Stokes and Navier-Stokes equations to validate the method and the resulting code. (author)
Wang, Wansheng; Chen, Long; Zhou, Jie
2015-01-01
A postprocessing technique for mixed finite element methods for the Cahn-Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on the coarser mesh, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures. PMID:27110063
Solving Dym equation using quartic B-spline and quartic trigonometric B-spline collocation methods
Anuar, Hanis Safirah Saiful; Mafazi, Nur Hidayah; Hamid, Nur Nadiah Abd; Majid, Ahmad Abd.; Azmi, Amirah
2017-08-01
The nonlinear Dym equation is solved numerically using the quartic B-spline (QuBS) and quartic trigonometric B-spline (QuTBS) collocation methods. The QuBS and QuTBS are utilized as interpolating functions in the spatial dimension while the finite difference method (FDM) is applied to discretize the temporal space with the help of theta-weighted method. The nonlinear term in the Dym equation is linearized using Taylor's expansion. Two schemes are performed on both methods which are Crank-Nicolson and fully implicit. Applying the Von-Neumann stability analysis, these schemes are found to be conditionally stable. Several numerical examples of different forms are discussed and compared in term of errors with exact solutions and results from the FDM.
Lid-driven cavity flow using a discrete velocity method for solving the Boltzmann equation
Sekaran, Aarthi; Varghese, Philip; Estes, Samuel; Goldstein, David
2016-11-01
We extend the discrete velocity method for solving the Boltzmann equation previously used for one-dimensional problems to two spatial dimensions. The collision integral is computed using collisions between velocity classes selected randomly using a Monte Carlo method. Arbitrary post-collision velocities are mapped back onto the grid using a projection scheme which conserves mass, momentum, and energy. In addition, a variance reduction scheme is implemented to decrease noise and further reduce computational effort. The convection part of the equation is computed using first order upwind finite differences. We apply this discrete velocity scheme to the 2D lid-driven square cavity flow problem with Ar as the fluid medium and explore the effect of the additional flexibility available in this quasi-particle based stochastic method on the accuracy and noise level in the solutions obtained.
An Archived firefly Algorithm; A mathematical software to solve univariate nonlinear equations
Directory of Open Access Journals (Sweden)
M.K.A. Ariyaratne
2016-06-01
Full Text Available In this article, we are presenting a software solution that proposes some modifications to the existing firefly algorithm. The modification; archived firefly algorithm [AFFA] exhibits the ability of finding almost all complex roots of a given nonlinear equation within a reasonable range. The software implementation includes two main properties; an archive to collect the better fireflies and a flag to determine poor performance in firefly generations. The new modification is tested over genetic algorithms (GA, a phenomenal success in the field of nature inspired algorithms and also with a modified GA embedded with same properties that the AFFA has. A simple graphical user interface (GUI is developed using MATLAB GUIDE to present the findings. Computer simulations show that the AFFA performs well in solving nonlinear equations with real as well as complex roots within a specified region.
CrasyDSE: A framework for solving Dyson-Schwinger equations
Huber, Markus Q.; Mitter, Mario
2012-11-01
Dyson-Schwinger equations are important tools for non-perturbative analyses of quantum field theories. For example, they are very useful for investigations in quantum chromodynamics and related theories. However, sometimes progress is impeded by the complexity of the equations. Thus automating parts of the calculations will certainly be helpful in future investigations. In this article we present a framework for such an automation based on a C++ code that can deal with a large number of Green functions. Since also the creation of the expressions for the integrals of the Dyson-Schwinger equations needs to be automated, we defer this task to a Mathematica notebook. We illustrate the complete workflow with an example from Yang-Mills theory coupled to a fundamental scalar field that has been investigated recently. As a second example we calculate the propagators of pure Yang-Mills theory. Our code can serve as a basis for many further investigations where the equations are too complicated to tackle by hand. It also can easily be combined with DoFun, a program for the derivation of Dyson-Schwinger equations.Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 49030 No. of bytes in distributed program, including test data, etc.: 303958 Distribution format: tar.gz Programming language: Mathematica 8 and higher, C++. Computer: All on which Mathematica and C++ are available. Operating system: All on which Mathematica and C++ are available (Windows, Unix, Mac OS). Classification: 11.1, 11.4, 11.5, 11.6. Nature of problem: Solve (large) systems of Dyson-Schwinger equations numerically. Solution method: Create C++ functions in Mathematica to be used for the numeric code in C++. This code uses structures to handle large numbers of Green functions. Unusual features: Provides a tool to convert Mathematica expressions into C++ expressions including conversion of function names
CrasyDSE: A framework for solving Dyson-Schwinger equations.
Huber, Markus Q; Mitter, Mario
2012-11-01
Dyson-Schwinger equations are important tools for non-perturbative analyses of quantum field theories. For example, they are very useful for investigations in quantum chromodynamics and related theories. However, sometimes progress is impeded by the complexity of the equations. Thus automating parts of the calculations will certainly be helpful in future investigations. In this article we present a framework for such an automation based on a C++ code that can deal with a large number of Green functions. Since also the creation of the expressions for the integrals of the Dyson-Schwinger equations needs to be automated, we defer this task to a Mathematica notebook. We illustrate the complete workflow with an example from Yang-Mills theory coupled to a fundamental scalar field that has been investigated recently. As a second example we calculate the propagators of pure Yang-Mills theory. Our code can serve as a basis for many further investigations where the equations are too complicated to tackle by hand. It also can easily be combined with DoFun , a program for the derivation of Dyson-Schwinger equations. Program title : CrasyDSE Catalogue identifier : AEMY _v1_0 Program summary URL : http://cpc.cs.qub.ac.uk/summaries/AEMY_v1_0.html Program obtainable from : CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions : Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc. : 49030 No. of bytes in distributed program, including test data, etc. : 303958 Distribution format : tar.gz Programming language : Mathematica 8 and higher, C++ . Computer : All on which Mathematica and C++ are available. Operating system : All on which Mathematica and C++ are available (Windows, Unix, Mac OS). Classification : 11.1, 11.4, 11.5, 11.6. Nature of problem : Solve (large) systems of Dyson-Schwinger equations numerically. Solution method : Create C++ functions in Mathematica to be used for
A new method for solving the MHD equations in the magnetosheath
Directory of Open Access Journals (Sweden)
C. Nabert
2013-03-01
Full Text Available We present a new analytical method to derive steady-state magnetohydrodynamic (MHD solutions of the magnetosheath in different levels of approximation. With this method, we calculate the magnetosheath's density, velocity, and magnetic field distribution as well as its geometry. Thereby, the solution depends on the geomagnetic dipole moment and solar wind conditions only. To simplify the representation, we restrict our model to northward IMF with the solar wind flow along the stagnation streamline. The sheath's geometry, with its boundaries, bow shock and magnetopause, is determined self-consistently. Our model is stationary and time relaxation has not to be considered as in global MHD simulations. Our method uses series expansion to transfer the MHD equations into a new set of ordinary differential equations. The number of equations is related to the level of approximation considered including different physical processes. These equations can be solved numerically; however, an analytical approach for the lowest-order approximation is also presented. This yields explicit expressions, not only for the flow and field variations but also for the magnetosheath thickness, depending on the solar wind parameters. Results are compared to THEMIS data and offer a detailed explanation of, e.g., the pile-up process and the corresponding plasma depletion layer, the bow shock and magnetopause geometry, the magnetosheath thickness, and the flow deceleration.
A novel algebraic procedure for solving non-linear evolution equations of higher order
International Nuclear Information System (INIS)
Huber, Alfred
2007-01-01
We report here a systematic approach that can easily be used for solving non-linear partial differential equations (nPDE), especially of higher order. We restrict the analysis to the so called evolution equations describing any wave propagation. The proposed new algebraic approach leads us to traveling wave solutions and moreover, new class of solution can be obtained. The crucial step of our method is the basic assumption that the solutions satisfy an ordinary differential equation (ODE) of first order that can be easily integrated. The validity and reliability of the method is tested by its application to some non-linear evolution equations. The important aspect of this paper however is the fact that we are able to calculate distinctive class of solutions which cannot be found in the current literature. In other words, using this new algebraic method the solution manifold is augmented to new class of solution functions. Simultaneously we would like to stress the necessity of such sophisticated methods since a general theory of nPDE does not exist. Otherwise, for practical use the algebraic construction of new class of solutions is of fundamental interest
Noniterative, unconditionally stable numerical techniques for solving condensational anddissolutional growth equations are given. Growth solutions are compared to Gear-code solutions forthree cases when growth is coupled to reversible equilibrium chemistry. In all cases, ...
A pertinent approach to solve nonlinear fuzzy integro-differential equations.
Narayanamoorthy, S; Sathiyapriya, S P
2016-01-01
Fuzzy integro-differential equations is one of the important parts of fuzzy analysis theory that holds theoretical as well as applicable values in analytical dynamics and so an appropriate computational algorithm to solve them is in essence. In this article, we use parametric forms of fuzzy numbers and suggest an applicable approach for solving nonlinear fuzzy integro-differential equations using homotopy perturbation method. A clear and detailed description of the proposed method is provided. Our main objective is to illustrate that the construction of appropriate convex homotopy in a proper way leads to highly accurate solutions with less computational work. The efficiency of the approximation technique is expressed via stability and convergence analysis so as to guarantee the efficiency and performance of the methodology. Numerical examples are demonstrated to verify the convergence and it reveals the validity of the presented numerical technique. Numerical results are tabulated and examined by comparing the obtained approximate solutions with the known exact solutions. Graphical representations of the exact and acquired approximate fuzzy solutions clarify the accuracy of the approach.
Solving delay differential equations in S-ADAPT by method of steps.
Bauer, Robert J; Mo, Gary; Krzyzanski, Wojciech
2013-09-01
S-ADAPT is a version of the ADAPT program that contains additional simulation and optimization abilities such as parametric population analysis. S-ADAPT utilizes LSODA to solve ordinary differential equations (ODEs), an algorithm designed for large dimension non-stiff and stiff problems. However, S-ADAPT does not have a solver for delay differential equations (DDEs). Our objective was to implement in S-ADAPT a DDE solver using the methods of steps. The method of steps allows one to solve virtually any DDE system by transforming it to an ODE system. The solver was validated for scalar linear DDEs with one delay and bolus and infusion inputs for which explicit analytic solutions were derived. Solutions of nonlinear DDE problems coded in S-ADAPT were validated by comparing them with ones obtained by the MATLAB DDE solver dde23. The estimation of parameters was tested on the MATLB simulated population pharmacodynamics data. The comparison of S-ADAPT generated solutions for DDE problems with the explicit solutions as well as MATLAB produced solutions which agreed to at least 7 significant digits. The population parameter estimates from using importance sampling expectation-maximization in S-ADAPT agreed with ones used to generate the data. Published by Elsevier Ireland Ltd.
Solving many-body Schrödinger equations with kinetic energy partition method
Chen, Yu-Hsin; Chao, Sheng D.
2018-01-01
We present a general formulation of our previously developed kinetic energy partition (KEP) method for solving many-bodySchrödinger equations. In atomic physics, as well as in general molecular and solid state physics, solving many-electronSchrödinger equations is a very challenging task, often called Dirac's challenge. The central problem is how to properly handle the electron-electron Coulomb repulsion interactions. Using the KEP solution scheme, in addition to dividing the kinetic energy into partial terms, the electron-electron Coulomb interaction is also separated into parts to be associated with a "negative mass" kinetic energy term. Therefore, the full Hamiltonian can be expressed as a simple sum of subsystem Hamiltonians, each representing an effective one-body problem. Using a Hartree-like product in constructing the wave-function, we achieve fast convergence in the calculations of the ground state energies. First, the model Moshinsky atoms are used to illustrate the solution procedure. We then apply this new KEP method to harmonium atoms and obtain precise energies with an error less than 5% using only two basis functions from each subsystem. It is thus very promising that this methodology, when further extended, can be useful for general many-body systems.
Yoon, Gangjoon; Min, Chohong
2017-11-01
The Shortley-Weller method is a standard finite difference method for solving the Poisson equation with Dirichlet boundary condition. Unless the domain is rectangular, the method meets an inevitable problem that some of the neighboring nodes may be outside the domain. In this case, an usual treatment is to extrapolate the function values at outside nodes by quadratic polynomial. The extrapolation may become unstable in the sense that some of the extrapolation coefficients increase rapidly when the grid nodes are getting closer to the boundary. A practical remedy, which we call artificial perturbation, is to treat grid nodes very near the boundary as boundary points. The aim of this paper is to reveal the adverse effects of the artificial perturbation on solving the linear system and the convergence of the solution. We show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues is an important factor in solving the linear system. Our analysis shows that the artificial perturbation results in a small enhancement of the eigenvalue ratio from O (1 / (h ṡhmin) to O (h-3) and triggers an oscillatory order of convergence. Instead, we suggest using Jacobi or ILU-type preconditioner on the matrix without applying the artificial perturbation. According to our analysis, the preconditioning not only reduces the eigenvalue ratio from O (1 / (h ṡhmin) to O (h-2), but also keeps the sharp second order convergence.
Algorithms to solve coupled systems of differential equations in terms of power series
International Nuclear Information System (INIS)
Ablinger, Jakob; Schneider, Carsten
2016-08-01
Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following we suppose that the unknown Feynman integrals can be given in power series representations, and that sufficiently many initial values of the integrals are given. Then there exist algorithms that decide constructively if the coefficients of their power series representations can be given within the class of nested sums over hypergeometric products. In this article we work out the calculation steps that solve this problem. First, we present a successful tactic that has been applied recently to challenging problems coming from massive 3-loop Feynman integrals. Here our main tool is to solve scalar linear recurrences within the class of nested sums over hypergeometric products. Second, we will present a new variation of this tactic which relies on more involved summation technologies but succeeds in reducing the problem to solve scalar recurrences with lower recurrence orders. The article works out the different challenges of this new tactic and demonstrates how they can be treated efficiently with our existing summation technologies.
King, Nathan D.; Ruuth, Steven J.
2017-05-01
Maps from a source manifold M to a target manifold N appear in liquid crystals, color image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems and partial differential equations (PDEs) that map between manifolds is introduced within this paper. Our approach, the closest point method for manifold mapping, reduces the problem of solving a constrained PDE between manifolds M and N to the simpler problems of solving a PDE on M and projecting to the closest points on N. In our approach, an embedding PDE is formulated in the embedding space using closest point representations of M and N. This enables the use of standard Cartesian numerics for general manifolds that are open or closed, with or without orientation, and of any codimension. An algorithm is presented for the important example of harmonic maps and generalized to a broader class of PDEs, which includes p-harmonic maps. Improved efficiency and robustness are observed in convergence studies relative to the level set embedding methods. Harmonic and p-harmonic maps are computed for a variety of numerical examples. In these examples, we denoise texture maps, diffuse random maps between general manifolds, and enhance color images.
Collery, Olivier; Guyader, Jean-Louis
2010-03-01
In the context of better understanding and predicting sound transmission through heterogeneous fluid-loaded aircraft structures, this paper presents a method of solving the vibroacoustic problem of plates. The present work considers fluid-structure coupling and is applied to simply supported rectangular plates excited mechanically. The proposed method is based on the minimization of the error of verification of the plate vibroacoustic equation of motion on a sample of points. From sampling comes an aliasing effect; this phenomenon is described and solved using a wavelet-based filter. The proposed approach is validated in presenting very accurate results of sound radiation immersed in heavy and light fluids. The fluid-structure interaction appears to be very well described avoiding time-consuming classical calculations of the modal radiation impedances. The focus is also put on different samplings to observe the aliasing effect. As perspectives sound radiation from a non-homogeneous plate is solved and compared with reference results proving all the power of this method.
A new modified conjugate gradient coefficient for solving system of linear equations
Hajar, N.; ‘Aini, N.; Shapiee, N.; Abidin, Z. Z.; Khadijah, W.; Rivaie, M.; Mamat, M.
2017-09-01
Conjugate gradient (CG) method is an evolution of computational method in solving unconstrained optimization problems. This approach is easy to implement due to its simplicity and has been proven to be effective in solving real-life application. Although this field has received copious amount of attentions in recent years, some of the new approaches of CG algorithm cannot surpass the efficiency of the previous versions. Therefore, in this paper, a new CG coefficient which retains the sufficient descent and global convergence properties of the original CG methods is proposed. This new CG is tested on a set of test functions under exact line search. Its performance is then compared to that of some of the well-known previous CG methods based on number of iterations and CPU time. The results show that the new CG algorithm has the best efficiency amongst all the methods tested. This paper also includes an application of the new CG algorithm for solving large system of linear equations
Energy Technology Data Exchange (ETDEWEB)
Pusa, M.; Leppaenen, J. [VTT Technical Research Centre of Finland, P.O. Box 1000, FI-02044 VTT (Finland)
2012-07-01
The Chebyshev Rational Approximation Method (CRAM) has been recently introduced by the authors for solving the burnup equations with excellent results. This method has been shown to be capable of simultaneously solving an entire burnup system with thousands of nuclides both accurately and efficiently. The method was prompted by an analysis of the spectral properties of burnup matrices and it can be characterized as the best rational approximation on the negative real axis. The coefficients of the rational approximation are fixed and have been reported for various approximation orders. In addition to these coefficients, implementing the method only requires a linear solver. This paper describes an efficient method for solving the linear systems associated with the CRAM approximation. The introduced direct method is based on sparse Gaussian elimination where the sparsity pattern of the resulting upper triangular matrix is determined before the numerical elimination phase. The stability of the proposed Gaussian elimination method is discussed based on considering the numerical properties of burnup matrices. Suitable algorithms are presented for computing the symbolic factorization and numerical elimination in order to facilitate the implementation of CRAM and its adoption into routine use. The accuracy and efficiency of the described technique are demonstrated by computing the CRAM approximations for a large test case with over 1600 nuclides. (authors)
A Time Marching Scheme for Solving Volume Integral Equations on Nonlinear Scatterers
Bagci, Hakan
2015-01-07
Transient electromagnetic field interactions on inhomogeneous penetrable scatterers can be analyzed by solving time domain volume integral equations (TDVIEs). TDVIEs are oftentimes solved using marchingon-in-time (MOT) schemes. Unlike finite difference and finite element schemes, MOT-TDVIE solvers require discretization of only the scatterers, do not call for artificial absorbing boundary conditions, and are more robust to numerical phase dispersion. On the other hand, their computational cost is high, they suffer from late-time instabilities, and their implicit nature makes incorporation of nonlinear constitutive relations more difficult. Development of plane-wave time-domain (PWTD) and FFT-based schemes has significantly reduced the computational cost of the MOT-TDVIE solvers. Additionally, latetime instability problem has been alleviated for all practical purposes with the development of accurate integration schemes and specially designed temporal basis functions. Addressing the third challenge is the topic of this presentation. I will talk about an explicit MOT scheme developed for solving the TDVIE on scatterers with nonlinear material properties. The proposed scheme separately discretizes the TDVIE and the nonlinear constitutive relation between electric field intensity and flux density. The unknown field intensity and flux density are expanded using half and full Schaubert-Wilton-Glisson (SWG) basis functions in space and polynomial temporal interpolators in time. The resulting coupled system of the discretized TDVIE and constitutive relation is integrated in time using an explicit P E(CE) m scheme to yield the unknown expansion coefficients. Explicitness of time marching allows for straightforward incorporation of the nonlinearity as a function evaluation on the right hand side of the coupled system of equations. Consequently, the resulting MOT scheme does not call for a Newton-like nonlinear solver. Numerical examples, which demonstrate the applicability
Dynamics of vortices in superconductors
International Nuclear Information System (INIS)
Weinan, E.
1992-01-01
We study the dynamics of vortices in type-II superconductors from the point of view of time-dependent Ginzburg-Landau equations. We outline a proof of existence, uniqueness and regularity of strong solutions for these equations. We then derive reduced systems of ODEs governing the motion of the vortices in the asymptotic limit of large Ginzburg-Landau parameter
A finite element multigrid-framework to solve the sea ice momentum equation
Mehlmann, C.; Richter, T.
2017-11-01
We present a multigrid framework for the solution of the momentum equation arising in Hibler's viscous-plastic (VP) sea ice model. This model is used in global climate models, for seasonal prediction and high-resolution short-term forecasting systems. The development of fast and robust solvers for the strongly nonlinear momentum equation is still a big issue. There are mainly three approaches to solve the momentum equation: a fixed-point iteration (Picard solver), an inexact Newton method and a subcycling procedure based on an elastic-viscous-plastic (EVP) model approximation. Simple fixed-point iterations call for a vast number of cycles. The Jacobian arising in Newton linearizations is ill-conditioned and unstructured. No efficient linear solvers are available up to date. One possibility to solve the linear systems arising within a Newton method is the preconditioned GMRES iteration that however still requires many steps. The commonly used line SOR preconditioner is computationally expensive. Especially on fine meshes, literature recommends to revise common solution strategies. We introduce a geometric multigrid method as a preconditioner to the GMRES iteration for accelerating the solution of the linear problems. We show that the convergence rate of the multigrid method is robust with respect to mesh refinement. This makes it an appealing method for high resolution simulations. We validate the robustness of the linear solver and compare the multigrid with ILU preconditioning. In particular on fine meshes (∼16 km-2 km), multigrid preconditioning can substantially reduce the computational effort and decreases iteration counts by 80%.
Mechanical Analogy-based Iterative Method for Solving a System of Linear Equations
Directory of Open Access Journals (Sweden)
Yu. V. Berchun
2015-01-01
Full Text Available The paper reviews prerequisites to creating a variety of the iterative methods to solve a system of linear equations (SLE. It considers the splitting methods, variation-type methods, projection-type methods, and the methods of relaxation.A new iterative method based on mechanical analogy (the movement without resistance of a material point, that is connected by ideal elastically-linear constraints with unending guides defined by equations of solved SLE. The mechanical system has the unique position of stable equilibrium, the coordinates of which correspond to the solution of linear algebraic equation. The model of the mechanical system is a system of ordinary differential equations of the second order, integration of which allows you to define the point trajectory. In contrast to the classical methods of relaxation the proposed method does not ensure a trajectory passage through the equilibrium position. Thus the convergence of the method is achieved through the iterative stop of a material point at the moment it passes through the next (from the beginning of the given iteration minimum of potential energy. After that the next iteration (with changed initial coordinates starts.A resource-intensive process of numerical integration of differential equations in order to obtain a precise law of motion (at each iteration is replaced by defining its approximation. The coefficients of the approximating polynomial of the fourth order are calculated from the initial conditions, including higher-order derivatives. The resulting approximation enables you to evaluate the kinetic energy of a material point to calculate approximately the moment of time to reach the maximum kinetic energy (and minimum of the potential one, i.e. the end of the iteration.The software implementation is done. The problems with symmetric positive definite matrix, generated as a result of using finite element method, allowed us to examine a convergence rate of the proposed method
Directory of Open Access Journals (Sweden)
Tsugio Fukuchi
2014-06-01
Full Text Available The finite difference method (FDM based on Cartesian coordinate systems can be applied to numerical analyses over any complex domain. A complex domain is usually taken to mean that the geometry of an immersed body in a fluid is complex; here, it means simply an analytical domain of arbitrary configuration. In such an approach, we do not need to treat the outer and inner boundaries differently in numerical calculations; both are treated in the same way. Using a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. Discretization of the space differential in the FDM is usually derived using the Taylor series expansion; however, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences. In using the polynomial interpolations, we can numerically solve the Poisson equation freely over any complex domain. Only a particular type of partial differential equation, Poisson's equations, is treated; however, the arguments put forward have wider generality in numerical calculations using the FDM.
Solving the Poisson partial differential equation using vector space projection methods
Marendic, Boris
This research presents a new approach at solving the Poisson partial differential equation using Vector Space Projection (VSP) methods. The work attacks the Poisson equation as encountered in two-dimensional phase unwrapping problems, and in two-dimensional electrostatic problems. Algorithms are developed by first considering simple one-dimensional cases, and then extending them to two-dimensional problems. In the context of phase unwrapping of two-dimensional phase functions, we explore an approach to the unwrapping using a robust extrapolation-projection algorithm. The unwrapping is done iteratively by modification of the Gerchberg-Papoulis (GP) extrapolation algorithm, and the solution is refined by projecting onto the available global data. An important contribution to the extrapolation algorithm is the formulation of the algorithm with the relaxed bandwidth constraint, and the proof that such modified GP extrapolation algorithm still converges. It is also shown that the unwrapping problem is ill-posed in the VSP setting, and that the modified GP algorithm is the missing link to pushing the iterative algorithm out of the trap solution under certain conditions. Robustness of the algorithm is demonstrated through its performance in a noisy environment. Performance is demonstrated by applying it to phantom phase functions, as well as to the real phase functions. Results are compared to well known algorithms in literature. Unlike many existing unwrapping methods which perform unwrapping locally, this work approaches the unwrapping problem from a globally, and eliminates the need for guiding instruments, like quality maps. VSP algorithm also very effectively battles problems of shadowing and holes, where data is not available or is heavily corrupted. In solving the classical Poisson problems in electrostatics, we demonstrate the effectiveness and ease of implementation of the VSP methodology to solving the equation, as well as imposing of the boundary conditions
Markovian Monte Carlo program EvolFMC v.2 for solving QCD evolution equations
Jadach, S.; Płaczek, W.; Skrzypek, M.; Stokłosa, P.
2010-02-01
We present the program EvolFMC v.2 that solves the evolution equations in QCD for the parton momentum distributions by means of the Monte Carlo technique based on the Markovian process. The program solves the DGLAP-type evolution as well as modified-DGLAP ones. In both cases the evolution can be performed in the LO or NLO approximation. The quarks are treated as massless. The overall technical precision of the code has been established at 5×10. This way, for the first time ever, we demonstrate that with the Monte Carlo method one can solve the evolution equations with precision comparable to the other numerical methods. New version program summaryProgram title: EvolFMC v.2 Catalogue identifier: AEFN_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFN_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including binary test data, etc.: 66 456 (7407 lines of C++ code) No. of bytes in distributed program, including test data, etc.: 412 752 Distribution format: tar.gz Programming language: C++ Computer: PC, Mac Operating system: Linux, Mac OS X RAM: Less than 256 MB Classification: 11.5 External routines: ROOT ( http://root.cern.ch/drupal/) Nature of problem: Solution of the QCD evolution equations for the parton momentum distributions of the DGLAP- and modified-DGLAP-type in the LO and NLO approximations. Solution method: Monte Carlo simulation of the Markovian process of a multiple emission of partons. Restrictions:Limited to the case of massless partons. Implemented in the LO and NLO approximations only. Weighted events only. Unusual features: Modified-DGLAP evolutions included up to the NLO level. Additional comments: Technical precision established at 5×10. Running time: For the 10 6 events at 100 GeV: DGLAP NLO: 27s; C-type modified DGLAP NLO: 150s (MacBook Pro with Mac OS X v.10
S. Saha Ray; S. Sahoo
2016-01-01
The present paper deals with two reliable efficient methods viz. tanh-sech method and modified Kudryashov method, which are used to solve time-fractional nonlinear evolution equation. For delineating the legitimacy of proposed methods, we employ it to the time-fractional fifth-order modified Sawada–Kotera equations. As a consequence, we effectively obtained more new exact solutions for time-fractional fifth-order modified Sawada–Kotera equation. We have also presented the numerical simulation...
Directory of Open Access Journals (Sweden)
Farshid Mirzaee
2014-06-01
Full Text Available In this paper, we present a numerical method for solving two-dimensional Fredholm–Volterra integral equations (F-VIE. The method reduces the solution of these integral equations to the solution of a linear system of algebraic equations. The existence and uniqueness of the solution and error analysis of proposed method are discussed. The method is computationally very simple and attractive. Finally, numerical examples illustrate the efficiency and accuracy of the method.
On the proximity effect in a superconductive slab bordered by metal
International Nuclear Information System (INIS)
Liniger, W.
1993-01-01
The first Ginzburg-Landau equation for the order parameter ψ in the absence of magnetic fields is solved analytically for a superconducting slab of thickness 2d boardered by semi-infinite regions of normal metal at each face. The real-valued normalized wave function f=ψ/ψ ∞ depends only on the transversal spatial coordinate x, normalized with respect to the coherence length ξ of the superconductor, provided the de Gennes boundary condition df/dx=f/b is used. The closed-form solution expresses x as an elliptic integral of f, depending on the normalized parameters d and b. It is predicted theoretically that, for b c =arctan(1/b), the proximity effect is so strong that the superconductivity is completely suppressed. In fact, in this case, the first Ginzburg-Landau equation possesses only the trivial solution f≡0
PARALLEL SMAC ALGORITHMS TO SOLVE SHALLOW WATER EQUATION WITH UNSTRUCTURED COLLOCATED GRID SYSTEM
Yamashita, Haruka; Ushijima, Satoru
A computational method to solve shallow water equations has been investigated with an SMAC method which is usually employed in the simulation for incompressible fluids. In particular, this numerical method is implemented in the unstructured collocated grid system with the distributed memory system to increase the parallel efficiency. The developed computational method was applied to the 1D dam-break problem and the free-surface flows in a meandering open channel. As a result of the 1D dam-break simulations, it was confirmed that this method improve the numerical stability. While, in the case of the meandering open channel, it was confirmed that the predicted water depth and depth-averaged velocity distributions are qualitatively in good agreement with the experimental results and that the reasonable parallel efficiencies are attained by parallel computations.
Diagnostic models of intelligent tutor system for teaching skills to solve algebraic equations
Directory of Open Access Journals (Sweden)
Andrey Grigoriyevich Chukhray
2007-10-01
Full Text Available In this paper one solution for teaching skills to solve n-power algebraic equation by Lobachevsky-Greffe-Dandelen method is described. StudentÃ¢Â€Â™s mistakes are discovered and classified. Based on signal-parametric approach to fault diagnosis in dynamic systems mathematical diagnostic models which allow detecting mistake classes by comparing student calculated results and system calculated results are created. Features of proposed diagnostic models application are presented. Intelligent tutor system is developed and used on Ã¢Â€ÂœAutomatic Control TheoryÃ¢Â€Â practical training by third year students of National Aerospace University.
International Nuclear Information System (INIS)
Tumelero, Fernanda; Petersen, Claudio Zen; Goncalves, Glenio Aguiar; Schramm, Marcelo
2016-01-01
In this work, we report a solution to solve the Neutron Point Kinetics Equations applying the Polynomial Approach Method. The main idea is to expand the neutron density and delayed neutron precursors as a power series considering the reactivity as an arbitrary function of the time in a relatively short time interval around an ordinary point. In the first interval one applies the initial conditions and the analytical continuation is used to determine the solutions of the next intervals. A genuine error control is developed based on an analogy with the Rest Theorem. For illustration, we also report simulations for different approaches types (linear, quadratic and cubic). The results obtained by numerical simulations for linear approximation are compared with results in the literature.
Program to solve the multigroup discrete ordinates transport equation in (x,y,z) geometry
International Nuclear Information System (INIS)
Lathrop, K.D.
1976-04-01
Numerical formulations and programming algorithms are given for the THREETRAN computer program which solves the discrete ordinates, multigroup transport equation in (x,y,z) geometry. An efficient, flexible, and general data-handling strategy is derived to make use of three hierarchies of storage: small core memory, large core memory, and disk file. Data management, input instructions, and sample problem output are described. A six-group, S 4 , 18 502 mesh point, 2 800 zone, k/sub eff/ calculation of the ZPPR-4 critical assembly required 144 min of CDC-7600 time to execute to a convergence tolerance of 5 x 10 -4 and gave results in good qualitative agreement with experiment and other calculations. 6 references
Energy Technology Data Exchange (ETDEWEB)
Tumelero, Fernanda; Petersen, Claudio Zen; Goncalves, Glenio Aguiar [Universidade Federal de Pelotas, Capao do Leao, RS (Brazil). Programa de Pos Graduacao em Modelagem Matematica; Schramm, Marcelo [Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica
2016-12-15
In this work, we report a solution to solve the Neutron Point Kinetics Equations applying the Polynomial Approach Method. The main idea is to expand the neutron density and delayed neutron precursors as a power series considering the reactivity as an arbitrary function of the time in a relatively short time interval around an ordinary point. In the first interval one applies the initial conditions and the analytical continuation is used to determine the solutions of the next intervals. A genuine error control is developed based on an analogy with the Rest Theorem. For illustration, we also report simulations for different approaches types (linear, quadratic and cubic). The results obtained by numerical simulations for linear approximation are compared with results in the literature.
A theory of solving TAP equations for Ising models with general invariant random matrices
Opper, Manfred; Çakmak, Burak; Winther, Ole
2016-03-01
We consider the problem of solving TAP mean field equations by iteration for Ising models with coupling matrices that are drawn at random from general invariant ensembles. We develop an analysis of iterative algorithms using a dynamical functional approach that in the thermodynamic limit yields an effective dynamics of a single variable trajectory. Our main novel contribution is the expression for the implicit memory term of the dynamics for general invariant ensembles. By subtracting these terms, that depend on magnetizations at previous time steps, the implicit memory terms cancel making the iteration dependent on a Gaussian distributed field only. The TAP magnetizations are stable fixed points if a de Almeida-Thouless stability criterion is fulfilled. We illustrate our method explicitly for coupling matrices drawn from the random orthogonal ensemble.
Iterative Method to Solve a Data Completion Problem for Biharmonic Equation for Rectangular Domain
Directory of Open Access Journals (Sweden)
Tajani Chakir
2017-07-01
Full Text Available In this work, we are interested in a class of problems of great importance in many areas of industry and engineering. It is the invese problem for the biharmonic equation. It consists to complete the missing data on the inaccessible part from the measured data on the accessible part of the boundary. To solve this ill-posed problem, we opted for the alternative iterative method developed by Kozlov, Mazya and Fomin which is a convergent method for the elliptical Cauchy problems in general. The numerical implementation of the iterative algorithm is based on the application of the boundary element method (BEM for a sequence of mixed well-posed direct problems. Numerical results are performed for a square domain showing the effectiveness of the algorithm by BEM to produce accurate and stable numerical results.
Energy Technology Data Exchange (ETDEWEB)
Carella, Alfredo Raul
2012-09-15
Quantifying species transport rates is a main concern in chemical and petrochemical industries. In particular, the design and operation of many large-scale industrial chemical processes is as much dependent on diffusion as it is on reaction rates. However, the existing diffusion models sometimes fail to predict experimentally observed behaviors and their accuracy is usually insufficient for process optimization purposes. Fractional diffusion models offer multiple possibilities for generalizing Flick's law in a consistent manner in order to account for history dependence and nonlocal effects. These models have not been extensively applied to the study of real systems, mainly due to their computational cost and mathematical complexity. A least squares spectral formulation was developed for solving fractional differential equations. The proposed method was proven particularly well-suited for dealing with the numerical difficulties inherent to fractional differential operators. The practical implementation was explained in detail in order to enhance reproducibility, and directions were specified for extending it to multiple dimensions and arbitrarily shaped domains. A numerical framework based on the least-squares spectral element method was developed for studying and comparing anomalous diffusion models in pellets. This simulation tool is capable of solving arbitrary integro-differential equations and can be effortlessly adapted to various problems in any number of dimensions. Simulations of the flow around a cylindrical particle were achieved by extending the functionality of the developed framework. A test case was analyzed by coupling the boundary condition yielded by the fluid model with two families of anomalous diffusion models: hyperbolic diffusion and fractional diffusion. Qualitative guidelines for determining the suitability of diffusion models can be formulated by complementing experimental data with the results obtained from this approach.(Author)
Accelerated procedure to solve kinetic equation for neutral atoms in a hot plasma
Tokar, Mikhail Z.
2017-12-01
The recombination of plasma charged components, electrons and ions of hydrogen isotopes, on the wall of a fusion reactor is a source of neutral molecules and atoms, recycling back into the plasma volume. Here neutral species participate, in particular, in charge-exchange (c-x) collisions with the plasma ions and, as a result, atoms of high energies with chaotically directed velocities are generated. Some fraction of these hot atoms hit the wall. Statistical Monte Carlo methods normally used to model c-x atoms are too time consuming for reasonably small level of accident errors and extensive parameter studies are problematic. By applying pass method to evaluate integrals from functions, including the ion velocity distribution, an iteration approach to solve one-dimensional kinetic equation [1], being alternative to Monte Carlo procedure, has been tremendously accelerated, at least by a factor of 30-50 [2]. Here this approach is developed further to solve the 2-D kinetic equation, applied to model the transport of c-x atoms in the vicinity of an opening in the wall, e.g., the entrance of the duct guiding to a diagnostic installation. This is necessary to determine firmly the energy spectrum of c-x atoms penetrating into the duct and to assess the erosion of the installation there. The results of kinetic modeling are compared with those obtained with the diffusion description for c-x atoms, being strictly relevant under plasma conditions of low temperature and high density, where the mean free path length between c-x collisions is much smaller than that till the atom ionization by electrons. It is demonstrated that the previous calculations [3], done with the diffusion approximation for c-x atoms, overestimate the erosion rate of Mo mirrors in a reactor by a factor of 3 compared to the result of the present kinetic study.
Eshkuvatov, Z K; Zulkarnain, F S; Nik Long, N M A; Muminov, Z
2016-01-01
Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [-1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707-1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265-274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636-641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.
pK(A) in proteins solving the Poisson-Boltzmann equation with finite elements.
Sakalli, Ilkay; Knapp, Ernst-Walter
2015-11-05
Knowledge on pK(A) values is an eminent factor to understand the function of proteins in living systems. We present a novel approach demonstrating that the finite element (FE) method of solving the linearized Poisson-Boltzmann equation (lPBE) can successfully be used to compute pK(A) values in proteins with high accuracy as a possible replacement to finite difference (FD) method. For this purpose, we implemented the software molecular Finite Element Solver (mFES) in the framework of the Karlsberg+ program to compute pK(A) values. This work focuses on a comparison between pK(A) computations obtained with the well-established FD method and with the new developed FE method mFES, solving the lPBE using protein crystal structures without conformational changes. Accurate and coarse model systems are set up with mFES using a similar number of unknowns compared with the FD method. Our FE method delivers results for computations of pK(A) values and interaction energies of titratable groups, which are comparable in accuracy. We introduce different thermodynamic cycles to evaluate pK(A) values and we show for the FE method how different parameters influence the accuracy of computed pK(A) values. © 2015 Wiley Periodicals, Inc.
Tahayori, Bahman; Johnston, Leigh A; Layton, Kelvin J; Farrell, Peter M; Mareels, Iven M Y
2015-10-01
In waveform design for magnetic resonance applications, periodic continuous-wave excitation offers potential advantages that remain largely unexplored because of a lack of understanding of the Bloch equation with periodic continuous-wave excitations. Using harmonic balancing techniques the steady state solutions of the Bloch equation with periodic excitation can be effectively solved. Moreover, the convergence speed of the proposed series approximation is such that a few terms in the series expansion suffice to obtain a very accurate description of the steady state solution. The accuracy of the proposed analytic approximate series solution is verified using both a simulation study as well as experimental data derived from a spherical phantom with doped water under continuous-wave excitation. Typically a five term series suffices to achieve a relative error of less than one percent, allowing for a very effective and efficient analytical design process. The opportunities for Rabi frequency modulated continuous-wave form excitation are then explored, based on a comparison with steady state free precession pulse sequences.
A numerical spectral approach to solve the dislocation density transport equation
International Nuclear Information System (INIS)
Djaka, K S; Taupin, V; Berbenni, S; Fressengeas, C
2015-01-01
A numerical spectral approach is developed to solve in a fast, stable and accurate fashion, the quasi-linear hyperbolic transport equation governing the spatio-temporal evolution of the dislocation density tensor in the mechanics of dislocation fields. The approach relies on using the Fast Fourier Transform algorithm. Low-pass spectral filters are employed to control both the high frequency Gibbs oscillations inherent to the Fourier method and the fast-growing numerical instabilities resulting from the hyperbolic nature of the transport equation. The numerical scheme is validated by comparison with an exact solution in the 1D case corresponding to dislocation dipole annihilation. The expansion and annihilation of dislocation loops in 2D and 3D settings are also produced and compared with finite element approximations. The spectral solutions are shown to be stable, more accurate for low Courant numbers and much less computation time-consuming than the finite element technique based on an explicit Galerkin-least squares scheme. (paper)
LOCFES-B: Solving the one-dimensional transport equation with user-selected spatial approximations
International Nuclear Information System (INIS)
Jarvis, R.D.; Nelson, P.
1993-01-01
Closed linear one-cell functional (CLOF) methods constitute an abstractly defined class of spatial approximations to the one-dimensional discrete ordinates equations of linear particle transport that encompass, as specific instances, the vast majority of the spatial approximations that have been either used or suggested in the computational solution of these equations. A specific instance of the class of CLOF methods is defined by a (typically small) number of functions of the cell width, total cross section, and direction cosine of particle motion. The LOCFES code takes advantage of the latter observation by permitting the use, within a more-or-less standard source iteration solution process, of an arbitrary CLOF method as defined by a user-supplied subroutine. The design objective of LOCFES was to provide automated determination of the order of accuracy (i.e., order of the discretization error) in the fine-mesh limit for an arbitrary user-selected CLOF method. This asymptotic order of accuracy is one widely used measure of the merit of a spatial approximation. This paper discusses LOCFES-B, which is a code that uses methods developed in LOCFES to solve one-dimensional linear particle transport problems with any user-selected CLOF method. LOCFES-B provides automatic solution of a given problem to within an accuracy specified by user input and provides comparison of the computational results against results from externally provided benchmark results
Convergence properties of iterative algorithms for solving the nodal diffusion equations
International Nuclear Information System (INIS)
Azmy, Y.Y.; Kirk, B.L.
1990-01-01
We drive the five point form of the nodal diffusion equations in two-dimensional Cartesian geometry and develop three iterative schemes to solve the discrete-variable equations: the unaccelerated, partial Successive Over Relaxation (SOR), and the full SOR methods. By decomposing the iteration error into its Fourier modes, we determine the spectral radius of each method for infinite medium, uniform model problems, and for the unaccelerated and partial SOR methods for finite medium, uniform model problems. Also for the two variants of the SOR method we determine the optimal relaxation factor that results in the smallest number of iterations required for convergence. Our results indicate that the number of iterations for the unaccelerated and partial SOR methods is second order in the number of nodes per dimension, while, for the full SOR this behavior is first order, resulting in much faster convergence for very large problems. We successfully verify the results of the spectral analysis against those of numerical experiments, and we show that for the full SOR method the linear dependence of the number of iterations on the number of nodes per dimension is relatively insensitive to the value of the relaxation parameter, and that it remains linear even for heterogenous problems. 14 refs., 1 fig
Solving the Bars-Green equation for moving mesons in two-dimensional QCD
Jia, Yu; Liang, Shuangran; Li, LiuJi; Xiong, Xiaonu
2017-11-01
The two-dimensional QCD in the large N limit, generally referred to as the 't Hooft model, is numerically investigated in the axial gauge in a comprehensive manner. The corresponding Bethe-Salpeter equation for a bound q\\overline{q} pair, originally derived by Bars and Green in 1978, was first numerically tackled by Li and collaborators in late 1980s, yet only for the stationary mesons. In this paper, we make further progress by numerically solving the Bars-Green equation for moving mesons, ranging from the chiral pion to charmonium. By choosing several different quark masses, we computed the corresponding quark condensates, meson spectra and their decay constants for a variety of meson momenta, and found satisfactory agreement with their counterparts obtained using light-cone gauge, thus numerically verifed the gauge and Poincaré invariance of the 't Hooft model. Moreover, we have explicitly confirmed that, as the meson gets more and more boosted, the large component of the Bars-Green wave function indeed approaches the corresponding 't Hooft light-cone wave function, while the small component of the wave function rapidly fades away.
An extended step characteristic method for solving the transport equation in general geometries
International Nuclear Information System (INIS)
DeHart, M.D.; Pevey, R.E.; Parish, T.A.
1994-01-01
A method for applying the discrete ordinates method to solve the Boltzmann transport equation on arbitrary two-dimensional meshes has been developed. The finite difference approach normally used to approximate spatial derivatives in extrapolating angular fluxes across a cell is replaced by direct solution of the characteristic form of the transport equation for each discrete direction. Thus, computational cells are not restricted to the geometrical shape of a mesh element characteristic of a given coordinate system. However, in terms of the treatment of energy and angular dependencies, this method resembles traditional discrete ordinates techniques. By using the method developed here, a general two-dimensional space can be approximated by an irregular mesh comprised of arbitrary polygons. Results for a number of test problems have been compared with solutions obtained from traditional methods, with good agreement. Comparisons include benchmarks against analytical results for problems with simple geometry, as well as numerical results obtained from traditional discrete ordinates methods by applying the ANISN and TWOTRAN-II computer programs
Solving the Bateman equations in CASMO5 using implicit ode numerical methods for stiff systems
Energy Technology Data Exchange (ETDEWEB)
Hykes, J. M.; Ferrer, R. M. [Studsvik Scandpower, Inc., 504 Shoup Avenue, Idaho Falls, ID (United States)
2013-07-01
The Bateman equations, which describe the transmutation of nuclides over time as a result of radioactive decay, absorption, and fission, are often numerically stiff. This is especially true if short-lived nuclides are included in the system. This paper describes the use of implicit numerical methods for o D Es applied to the stiff Bateman equations, specifically employing the Backward Differentiation Formulas (BDF) form of the linear multistep method. As is true in other domains, using an implicit method removes or lessens the (sometimes severe) step-length constraints by which explicit methods must abide. To gauge its accuracy and speed, the BDF method is compared to a variety of other solution methods, including Runge-Kutta explicit methods and matrix exponential methods such as the Chebyshev Rational Approximation Method (CRAM). A preliminary test case was chosen as representative of a PWR lattice depletion step and was solved with numerical libraries called from a Python front-end. The Figure of Merit (a combined measure of accuracy and efficiency) for the BDF method was nearly identical to that for CRAM, while explicit methods and other matrix exponential approximations trailed behind. The test case includes 319 nuclides, in which the shortest-lived nuclide is {sup 98}Nb with a half-life of 2.86 seconds. Finally, the BDF and CRAM methods were compared within CASMO5, where CRAM had a FOM about four times better than BDF, although the BDF implementation was not fully optimized. (authors)
A simple method for solving the Bussian equation for electrical conduction in rocks
Directory of Open Access Journals (Sweden)
P. W. J. Glover
2010-09-01
Full Text Available One of the most general and effective models for calculating the complex electrical conductivity and relative dielectric permittivity of rocks saturated with pore fluids is that of Bussian. Unlike most models, it is non-linear and cannot be solved algebraically. Consequently, researchers use reiterating numerical routines to obtain a solution of the equation, and then only for the real part of the solution. Here we present a different approach to the solution that uses conformal mapping in the complex plane, and implements it within Maple^{TM}. The method is simple and elegant in that it requires, for example, only 3 lines of code in Maple^{TM} 11 and little programming experience. The approach has been shown to be as precise as using the classical reiterating bisection method for real data implemented in C++ on an ordinary desktop computer to within a probability over 1 in 10^{9}. However, the conformal mapping approach is 52 times as fast. We show once more that the Bussian equation breaks down for low fluid conductivities, but recommend it (with the modified Archie's law for use with rocks saturated with high salinity fluids when the matrix is conductive.
Scilab software as an alternative low-cost computing in solving the linear equations problem
Agus, Fahrul; Haviluddin
2017-02-01
Numerical computation packages are widely used both in teaching and research. These packages consist of license (proprietary) and open source software (non-proprietary). One of the reasons to use the package is a complexity of mathematics function (i.e., linear problems). Also, number of variables in a linear or non-linear function has been increased. The aim of this paper was to reflect on key aspects related to the method, didactics and creative praxis in the teaching of linear equations in higher education. If implemented, it could be contribute to a better learning in mathematics area (i.e., solving simultaneous linear equations) that essential for future engineers. The focus of this study was to introduce an additional numerical computation package of Scilab as an alternative low-cost computing programming. In this paper, Scilab software was proposed some activities that related to the mathematical models. In this experiment, four numerical methods such as Gaussian Elimination, Gauss-Jordan, Inverse Matrix, and Lower-Upper Decomposition (LU) have been implemented. The results of this study showed that a routine or procedure in numerical methods have been created and explored by using Scilab procedures. Then, the routine of numerical method that could be as a teaching material course has exploited.
Abdallah, Ayman A.; Barnett, Alan R.; Ibrahim, Omar M.; Manella, Richard T.
1993-01-01
Within the MSC/NASTRAN DMAP (Direct Matrix Abstraction Program) module TRD1, solving physical (coupled) or modal (uncoupled) transient equations of motion is performed using the Newmark-Beta or mode superposition algorithms, respectively. For equations of motion with initial conditions, only the Newmark-Beta integration routine has been available in MSC/NASTRAN solution sequences for solving physical systems and in custom DMAP sequences or alters for solving modal systems. In some cases, one difficulty with using the Newmark-Beta method is that the process of selecting suitable integration time steps for obtaining acceptable results is lengthy. In addition, when very small step sizes are required, a large amount of time can be spent integrating the equations of motion. For certain aerospace applications, a significant time savings can be realized when the equations of motion are solved using an exact integration routine instead of the Newmark-Beta numerical algorithm. In order to solve modal equations of motion with initial conditions and take advantage of efficiencies gained when using uncoupled solution algorithms (like that within TRD1), an exact mode superposition method using MSC/NASTRAN DMAP has been developed and successfully implemented as an enhancement to an existing coupled loads methodology at the NASA Lewis Research Center.
Abdallah, Ayman A.; Barnett, Alan R.; Ibrahim, Omar M.; Manella, Richard T.
1993-05-01
Within the MSC/NASTRAN DMAP (Direct Matrix Abstraction Program) module TRD1, solving physical (coupled) or modal (uncoupled) transient equations of motion is performed using the Newmark-Beta or mode superposition algorithms, respectively. For equations of motion with initial conditions, only the Newmark-Beta integration routine has been available in MSC/NASTRAN solution sequences for solving physical systems and in custom DMAP sequences or alters for solving modal systems. In some cases, one difficulty with using the Newmark-Beta method is that the process of selecting suitable integration time steps for obtaining acceptable results is lengthy. In addition, when very small step sizes are required, a large amount of time can be spent integrating the equations of motion. For certain aerospace applications, a significant time savings can be realized when the equations of motion are solved using an exact integration routine instead of the Newmark-Beta numerical algorithm. In order to solve modal equations of motion with initial conditions and take advantage of efficiencies gained when using uncoupled solution algorithms (like that within TRD1), an exact mode superposition method using MSC/NASTRAN DMAP has been developed and successfully implemented as an enhancement to an existing coupled loads methodology at the NASA Lewis Research Center.
Second order method for solving 3D elasticity equations with complex interfaces
Wang, Bao; Xia, Kelin; Wei, Guo-Wei
2015-08-01
Elastic materials are ubiquitous in nature and indispensable components in man-made devices and equipments. When a device or equipment involves composite or multiple elastic materials, elasticity interface problems come into play. The solution of three-dimensional (3D) elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equations. This work introduces the matched interface and boundary (MIB) method for solving 3D elasticity interface problems. The proposed MIB elasticity interface scheme utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface. The interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines. Such an enforcement determines the fictitious values. A number of new techniques have been developed to construct efficient MIB elasticity interface schemes for dealing with cross derivative in coupled governing equations. The proposed method is extensively validated over both weak and strong discontinuities of the solution, both piecewise constant and position-dependent material parameters, both smooth and nonsmooth interface geometries, and both small and large contrasts in the Poisson's ratio and shear modulus across the interface. Numerical experiments indicate that the present MIB method is of second order convergence in both L∞ and L2 error norms for handling arbitrarily complex interfaces, including biomolecular surfaces. To our best knowledge, this is the first elasticity interface method that is able to deliver the second convergence for the molecular surfaces of proteins.
An Explicit Upwind Algorithm for Solving the Parabolized Navier-Stokes Equations
Korte, John J.
1991-01-01
An explicit, upwind algorithm was developed for the direct (noniterative) integration of the 3-D Parabolized Navier-Stokes (PNS) equations in a generalized coordinate system. The new algorithm uses upwind approximations of the numerical fluxes for the pressure and convection terms obtained by combining flux difference splittings (FDS) formed from the solution of an approximate Riemann (RP). The approximate RP is solved using an extension of the method developed by Roe for steady supersonic flow of an ideal gas. Roe's method is extended for use with the 3-D PNS equations expressed in generalized coordinates and to include Vigneron's technique of splitting the streamwise pressure gradient. The difficulty associated with applying Roe's scheme in the subsonic region is overcome. The second-order upwind differencing of the flux derivatives are obtained by adding FDS to either an original forward or backward differencing of the flux derivative. This approach is used to modify an explicit MacCormack differencing scheme into an upwind differencing scheme. The second order upwind flux approximations, applied with flux limiters, provide a method for numerically capturing shocks without the need for additional artificial damping terms which require adjustment by the user. In addition, a cubic equation is derived for determining Vegneron's pressure splitting coefficient using the updated streamwise flux vector. Decoding the streamwise flux vector with the updated value of Vigneron's pressure splitting improves the stability of the scheme. The new algorithm is applied to 2-D and 3-D supersonic and hypersonic laminar flow test cases. Results are presented for the experimental studies of Holden and of Tracy. In addition, a flow field solution is presented for a generic hypersonic aircraft at a Mach number of 24.5 and angle of attack of 1 degree. The computed results compare well to both experimental data and numerical results from other algorithms. Computational times required
Analysis and development of spatial hp-refinement methods for solving the neutron transport equation
International Nuclear Information System (INIS)
Fournier, D.
2011-01-01
The different neutronic parameters have to be calculated with a higher accuracy in order to design the 4. generation reactor cores. As memory storage and computation time are limited, adaptive methods are a solution to solve the neutron transport equation. The neutronic flux, solution of this equation, depends on the energy, angle and space. The different variables are successively discretized. The energy with a multigroup approach, considering the different quantities to be constant on each group, the angle by a collocation method called SN approximation. Once the energy and angle variable are discretized, a system of spatially-dependent hyperbolic equations has to be solved. Discontinuous finite elements are used to make possible the development of hp-refinement methods. Thus, the accuracy of the solution can be improved by spatial refinement (h-refinement), consisting into subdividing a cell into sub-cells, or by order refinement (p-refinement), by increasing the order of the polynomial basis. In this thesis, the properties of this methods are analyzed showing the importance of the regularity of the solution to choose the type of refinement. Thus, two error estimators are used to lead the refinement process. Whereas the first one requires high regularity hypothesis (analytical solution), the second one supposes only the minimal hypothesis required for the solution to exist. The comparison of both estimators is done on benchmarks where the analytic solution is known by the method of manufactured solutions. Thus, the behaviour of the solution as a regard of the regularity can be studied. It leads to a hp-refinement method using the two estimators. Then, a comparison is done with other existing methods on simplified but also realistic benchmarks coming from nuclear cores. These adaptive methods considerably reduces the computational cost and memory footprint. To further improve these two points, an approach with energy-dependent meshes is proposed. Actually, as the
Finite element discretization of non-linear diffusion equations with thermal fluctuations.
de la Torre, J A; Español, Pep; Donev, Aleksandar
2015-03-07
We present a finite element discretization of a non-linear diffusion equation used in the field of critical phenomena and, more recently, in the context of dynamic density functional theory. The discretized equation preserves the structure of the continuum equation. Specifically, it conserves the total number of particles and fulfills an H-theorem as the original partial differential equation. The discretization proposed suggests a particular definition of the discrete hydrodynamic variables in microscopic terms. These variables are then used to obtain, with the theory of coarse-graining, their dynamic equations for both averages and fluctuations. The hydrodynamic variables defined in this way lead to microscopically derived hydrodynamic equations that have a natural interpretation in terms of discretization of continuum equations. Also, the theory of coarse-graining allows to discuss the introduction of thermal fluctuations in a physically sensible way. The methodology proposed for the introduction of thermal fluctuations in finite element methods is general and valid for both regular and irregular grids in arbitrary dimensions. We focus here on simulations of the Ginzburg-Landau free energy functional using both regular and irregular 1D grids. Convergence of the numerical results is obtained for the static and dynamic structure factors as the resolution of the grid is increased.
International Nuclear Information System (INIS)
Angstmann, C.N.; Donnelly, I.C.; Henry, B.I.; Jacobs, B.A.; Langlands, T.A.M.; Nichols, J.A.
2016-01-01
We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also show that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.
Improvement of the Exp-function method for solving the BBM equation with time-dependent coefficients
Jahani, Maghsoud; Manafian, Jalil
2016-03-01
In this article, we establish the exact solutions for the BBM equation with time-dependent coefficients. The Exp-function method (EFM) and improvement of the Exp-function method (IEFM) are used to construct solitary and soliton solutions of nonlinear evolution equations. These methods are developed for searching exact travelling wave solutions of nonlinear partial differential equations. The exact particular solutions are of four types: the hyperbolic function solution, trigonometric function solution, exponential solution and rational solution. It is shown that the EFM and IEFM, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.
Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.
2012-10-01
We present a set of software routines in Maple 14 for solving first order ordinary differential equations (FOODEs). The package implements the Prelle-Singer method in its original form together with its extension to include integrating factors in terms of elementary functions. The package also presents a theoretical extension to deal with all FOODEs presenting Liouvillian solutions. Applications to ODEs taken from standard references show that it solves ODEs which remain unsolved using Maple's standard ODE solution routines. New version program summary Program title: PSsolver Catalogue identifier: ADPR_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADPR_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2302 No. of bytes in distributed program, including test data, etc.: 31962 Distribution format: tar.gz Programming language: Maple 14 (also tested using Maple 15 and 16). Computer: Intel Pentium Processor P6000, 1.86 GHz. Operating system: Windows 7. RAM: 4 GB DDR3 Memory Classification: 4.3. Catalogue identifier of previous version: ADPR_v1_0 Journal reference of previous version: Comput. Phys. Comm. 144 (2002) 46 Does the new version supersede the previous version?: Yes Nature of problem: Symbolic solution of first order differential equations via the Prelle-Singer method. Solution method: The method of solution is based on the standard Prelle-Singer method, with extensions for the cases when the FOODE contains elementary functions. Additionally, an extension of our own which solves FOODEs with Liouvillian solutions is included. Reasons for new version: The program was not running anymore due to changes in the latest versions of Maple. Additionally, we corrected/changed some bugs/details that were hampering the smoother functioning of the routines. Summary
Nakatsuji, Hiroshi; Nakashima, Hiroyuki
2015-02-01
The free-complement (FC) method is a general method for solving the Schrödinger equation (SE): The produced wave function has the potentially exact structure as the solution of the Schrödinger equation. The variables included are determined either by using the variational principle (FC-VP) or by imposing the local Schrödinger equations (FC-LSE) at the chosen set of the sampling points. The latter method, referred to as the local Schrödinger equation (LSE) method, is integral-free and therefore applicable to any atom and molecule. The purpose of this paper is to formulate the basic theories of the LSE method and explain their basic features. First, we formulate three variants of the LSE method, the AB, HS, and HTQ methods, and explain their properties. Then, the natures of the LSE methods are clarified in some detail using the simple examples of the hydrogen atom and the Hooke's atom. Finally, the ideas obtained in this study are applied to solving the SE of the helium atom highly accurately with the FC-LSE method. The results are very encouraging: we could get the world's most accurate energy of the helium atom within the sampling-type methodologies, which is comparable to those obtained with the FC-VP method. Thus, the FC-LSE method is an easy and yet a powerful integral-free method for solving the Schrödinger equation of general atoms and molecules.
Liu, Yang
2013-07-01
The computational complexity and memory requirements of multilevel plane wave time domain (PWTD)-accelerated marching-on-in-time (MOT)-based surface integral equation (SIE) solvers scale as O(NtNs(log 2)Ns) and O(Ns 1.5); here N t and Ns denote numbers of temporal and spatial basis functions discretizing the current [Shanker et al., IEEE Trans. Antennas Propag., 51, 628-641, 2003]. In the past, serial versions of these solvers have been successfully applied to the analysis of scattering from perfect electrically conducting as well as homogeneous penetrable targets involving up to Ns ≈ 0.5 × 106 and Nt ≈ 10 3. To solve larger problems, parallel PWTD-enhanced MOT solvers are called for. Even though a simple parallelization strategy was demonstrated in the context of electromagnetic compatibility analysis [M. Lu et al., in Proc. IEEE Int. Symp. AP-S, 4, 4212-4215, 2004], by and large, progress in this area has been slow. The lack of progress can be attributed wholesale to difficulties associated with the construction of a scalable PWTD kernel. © 2013 IEEE.
A Kronecker product variant of the FACR method for solving the generalized Poisson equation
Hendrickx, Jef; van Barel, Marc
2002-03-01
We present a fast direct method for the solution of a linear system , where M is a block tridiagonal Toeplitzmatrix with A on the diagonal and T on the two subdiagonals (A and T commute). Such matrices are obtained from a finite difference approximation to Poisson's equation with nonconstant coefficients in one direction (among others). The new method is called KPCR(l)-method and begins with l steps of cyclic reduction after which the remaining system is solved by a Kronecker product method. For an appropriate choice of l the asymptotic operation count for an n×n grid is O(n2 log2 log2 n), which is faster than either cyclic reduction or the Kronecker product method itself. The algorithm is similar to and has the same complexity as the FACR(l)-algorithm, which is a combination of cyclic reduction and Fourier analysis (or matrix decomposition). However, the FACR(l)-algorithm only reaches this complexity if A (and T) can be diagonalized by a fast transformation, where the new method is fast for every banded A and T. Moreover, the KPCR(l)-method can be easily generalized to the case where A and T do not commute.
Brehm, Christoph; Barad, Michael F.; Kiris, Cetin C.
2016-01-01
An immersed boundary method for the compressible Navier-Stokes equation and the additional infrastructure that is needed to solve moving boundary problems and fully coupled fluid-structure interaction is described. All the methods described in this paper were implemented in NASA's LAVA solver framework. The underlying immersed boundary method is based on the locally stabilized immersed boundary method that was previously introduced by the authors. In the present paper this method is extended to account for all aspects that are involved for fluid structure interaction simulations, such as fast geometry queries and stencil computations, the treatment of freshly cleared cells, and the coupling of the computational fluid dynamics solver with a linear structural finite element method. The current approach is validated for moving boundary problems with prescribed body motion and fully coupled fluid structure interaction problems in 2D and 3D. As part of the validation procedure, results from the second AIAA aeroelastic prediction workshop are also presented. The current paper is regarded as a proof of concept study, while more advanced methods for fluid structure interaction are currently being investigated, such as geometric and material nonlinearities, and advanced coupling approaches.
Predoi, Mihai Valentin
2014-09-01
The dispersion curves for hollow multilayered cylinders are prerequisites in any practical guided waves application on such structures. The equations for homogeneous isotropic materials have been established more than 120 years ago. The difficulties in finding numerical solutions to analytic expressions remain considerable, especially if the materials are orthotropic visco-elastic as in the composites used for pipes in the last decades. Among other numerical techniques, the semi-analytical finite elements method has proven its capability of solving this problem. Two possibilities exist to model a finite elements eigenvalue problem: a two-dimensional cross-section model of the pipe or a radial segment model, intersecting the layers between the inner and the outer radius of the pipe. The last possibility is here adopted and distinct differential problems are deduced for longitudinal L(0,n), torsional T(0,n) and flexural F(m,n) modes. Eigenvalue problems are deduced for the three modes classes, offering explicit forms of each coefficient for the matrices used in an available general purpose finite elements code. Comparisons with existing solutions for pipes filled with non-linear viscoelastic fluid or visco-elastic coatings as well as for a fully orthotropic hollow cylinder are all proving the reliability and ease of use of this method. Copyright © 2014 Elsevier B.V. All rights reserved.
Linear differential equations to solve nonlinear mechanical problems: A novel approach
Nair, C. Radhakrishnan
2004-01-01
Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential equation is known. Using the known solution of the non-linear differential equation, linear differential equations are set up. The solutions of these linear differential equations are found using standard techniques. Then the solutions of the linear differe...
Fonger, Nicole L.; Davis, Jon D.; Rohwer, Mary Lou
2018-01-01
This research addresses the issue of how to support students' representational fluency--the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is…
Saleem, M. Rehan; Ali, Ishtiaq; Qamar, Shamsul
2018-03-01
In this article, a reduced five-equation two-phase flow model is numerically investigated. The formulation of the model is based on the conservation and energy exchange laws. The model is non-conservative and the governing equations contain two equations for the mass conservation, one for the over all momentum and one for the total energy. The fifth equation is the energy equation for one of the two phases that includes a source term on the right hand side for incorporating energy exchange between the two fluids in the form of mechanical and thermodynamical works. A Runge-Kutta discontinuous Galerkin finite element method is applied to solve the model equations. The main attractive features of the proposed method include its formal higher order accuracy, its nonlinear stability, its ability to handle complicated geometries, and its ability to capture sharp discontinuities or strong gradients in the solutions without producing spurious oscillations. The proposed method is robust and well suited for large-scale time-dependent computational problems. Several case studies of two-phase flows are presented. For validation and comparison of the results, the same model equations are also solved by using a staggered central scheme. It was found that discontinuous Galerkin scheme produces better results as compared to the staggered central scheme.
Directory of Open Access Journals (Sweden)
M. Rehan Saleem
2018-03-01
Full Text Available In this article, a reduced five-equation two-phase flow model is numerically investigated. The formulation of the model is based on the conservation and energy exchange laws. The model is non-conservative and the governing equations contain two equations for the mass conservation, one for the over all momentum and one for the total energy. The fifth equation is the energy equation for one of the two phases that includes a source term on the right hand side for incorporating energy exchange between the two fluids in the form of mechanical and thermodynamical works. A Runge-Kutta discontinuous Galerkin finite element method is applied to solve the model equations. The main attractive features of the proposed method include its formal higher order accuracy, its nonlinear stability, its ability to handle complicated geometries, and its ability to capture sharp discontinuities or strong gradients in the solutions without producing spurious oscillations. The proposed method is robust and well suited for large-scale time-dependent computational problems. Several case studies of two-phase flows are presented. For validation and comparison of the results, the same model equations are also solved by using a staggered central scheme. It was found that discontinuous Galerkin scheme produces better results as compared to the staggered central scheme. Keywords: Two-phase compressible flows, Non-conservative system, Shock discontinuities, Discontinuous Galerkin method, Central scheme
Energy Technology Data Exchange (ETDEWEB)
Livio, Mario [Hubble Space Telescope Science Institute
2006-09-06
For thousands of years, mathematicians solved progressively more difficult algebraic equations, from the simple quadractic to the more complex quartic equation, yielding important insights along the way. Then they were stumped by the quintic equation, which resisted solutions for three centuries, until two great prodigies independently proved that quaintic equations cannot be solved by simple formula. These geniuses, a young Norwegian named Niels Henrik Abel and an even younger Frenchman named Evariste Galois, both died tragically. Galois' work gave rise to group theory, the "language" that defines symmetry. Group theory explains much about the aesthetics of our world, from the choosing of mates to Rubik's cube, Bach's musical compositions, the physics of subatomic particles, and the popularity of Anna Kournikova. Some of the mysteries surrounding Galois' death, which have lingered for more than 170 years, are finally resolved in The Equation that Couldn't Be Solved. Livio will discuss this first popular-level book to explore group theory, not through abstract formulas but in a dramatic account of the lives and work of some of the greatest mathematicians in history.
Methods and Algorithms for Solving Inverse Problems for Fractional Advection-Dispersion Equations
Aldoghaither, Abeer
2015-11-12
Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and long-rest periods. Mathematical modeling of physical phenomena requires the identification of pa- rameters and variables from available measurements. This is referred to as an inverse problem. In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional Advection-Dispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa- tion is important to understand how chemical or biological contaminants are trans- ported throughout surface aquifer systems. For instance, an estimate of the di↵eren- tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium. Our main contribution is to propose a novel e cient algorithm based on modulat-ing functions to estimate the coe cients and the di↵erentiation order for space FADE, which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show how the method can be applied to the source inverse problem. This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed two-stage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE. In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final
Energy Technology Data Exchange (ETDEWEB)
Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir [Faculty of Mathematics, Yazd University, Yazd (Iran, Islamic Republic of); The Laboratory of Quantum Information Processing, Yazd University, Yazd (Iran, Islamic Republic of); Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir [Faculty of Mathematics, Yazd University, Yazd (Iran, Islamic Republic of); The Laboratory of Quantum Information Processing, Yazd University, Yazd (Iran, Islamic Republic of); Cattani, C., E-mail: ccattani@unisa.it [Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano (Italy); Maalek Ghaini, F.M., E-mail: maalek@yazd.ac.ir [Faculty of Mathematics, Yazd University, Yazd (Iran, Islamic Republic of); The Laboratory of Quantum Information Processing, Yazd University, Yazd (Iran, Islamic Republic of)
2015-02-15
Because of the nonlinearity, closed-form solutions of many important stochastic functional equations are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. In this paper, a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of Itô-integration is proposed for solving nonlinear stochastic Itô integral equations in large intervals. In the proposed method, a new technique for computing nonlinear terms in such problems is presented. The main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved. Error analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As two useful applications, the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem.
A least-squares finite-element Sn method for solving first-order neutron transport equation
International Nuclear Information System (INIS)
Ju Haitao; Wu Hongchun; Zhou Yongqiang; Cao Liangzhi; Yao Dong; Xian, Chun-Yu
2007-01-01
A discrete ordinates finite-element method for solving the two-dimensional first-order neutron transport equation is derived using the least-squares variation. It avoids the singularity in void regions of the method derived from the second-order equation which contains the inversion of the cross-section. Different from using the standard Galerkin variation to the first-order equation, the least-squares variation results in a symmetric matrix, which can be solved easily and effectively. To eliminate the discontinuity of the angular flux on the vacuum boundary in the spherical harmonics method, the angle variable is discretized by the discrete ordinates method. A two-dimensional transport simulation code is developed and applied to some benchmark problems with unstructured geometry. The numerical results verified the validity of this method
Spectral difference methods for solving the differential equations of chemical physics
Mazziotti, David A.
2002-08-01
Spectral differences [D. A. Mazziotti, Chem. Phys. Lett. 299, 473 (1999)] is a family of techniques for solving differential equations in which the summation in the numerical derivative is accelerated to produce a matrix representation that is not only exponentially convergent like the discrete variable representation (DVR) and other spectral methods but also sparse like traditional finite differences and finite elements. Building upon important work by Boyd [Comput. Methods Appl. Mech. Eng. 116, 1 (1994)] and Gray and Goldfield [J. Chem. Phys. 115, 8331 (2001)], we explore a new class of spectral difference methods which yields solutions that are more accurate than high-order finite differences by several orders of magnitude. With the generating weight for Gegenbauer polynomials we design a new spectral difference method where the limits of an adjustable parameter alpha generate both finite differences (alpha=infinity), emphasizing the low Fourier frequencies, and a truncated sinc-DVR (alpha=0), emphasizing all Fourier frequencies below the aliasing limit of the grid. A range of choices for alpha[set membership]0,infinity produces solutions which are significantly better than the equivalent order of finite differences. We compare the Gegenbauer-weighted spectral differences with methods by Boyd as well as Gray and Goldfield which employ a hyperbolic secant and a step function as frequency weights, respectively. The solutions from the Gegenbauer- and the sech-weighted differences are shown to be less sensitive to parameter selection than the step-weighted differences. We illustrate all of the spectral difference methods through vibrational and quantum control calculations with diatomic iodine and the van der Waals cluster NeCO. Spectral differences also have important applications in molecular dynamics and electronic structure as well as other areas of science and engineering.
Directory of Open Access Journals (Sweden)
Yaoyu Hu
2015-09-01
Full Text Available The solution of the energy equation of thermo-elasto-hydrodynamic analysis for bearings by the finite element method usually leads to convergence difficulties due to the presence of convection terms inherited from the Navier–Stokes equations. In this work, the numerical analysis is performed with finite element method universally by adopting the characteristic-based split method to solve the energy equation. Five case studies of fixed pad thrust bearings have been set up with different geometries, loads, and lubricants. The two-dimensional film pressure is obtained by solving the Reynolds equation with pre-defined axial load on the pad. The energy equation of the lubricant film and the heat transfer equation of the bearing pad are handled by characteristic-based split method and conventional finite element method in three-dimensional space, respectively. Hot oil carry-over effect and variable lubricant viscosity are considered in the simulations. The results of the temperature distributions in the lubricant film and the bearing pad are presented. The possible usability of characteristic-based split method for future thermo-elasto-hydrodynamic analysis is discussed.
An efficient algorithm for solving nonlinear system of differential equations and applications
Directory of Open Access Journals (Sweden)
Mustafa GÜLSU
2015-10-01
Full Text Available In this article, we apply Chebyshev collocation method to obtain the numerical solutions of nonlinear systems of differential equations. This method transforms the nonlinear systems of differential equation to nonlinear systems of algebraic equations. The convergence of the numerical method are given and their applicability is illustrated with some examples.
International Nuclear Information System (INIS)
Núñez, Jóse; Ramos, Eduardo; Lopez, Juan M
2012-01-01
We describe a hybrid method based on the combined use of the Fourier Galerkin and finite-volume techniques to solve the fluid dynamics equations in cylindrical geometries. A Fourier expansion is used in the angular direction, partially translating the problem to the Fourier space and then solving the resulting equations using a finite-volume technique. We also describe an algorithm required to solve the coupled mass and momentum conservation equations similar to a pressure-correction SIMPLE method that is adapted for the present formulation. Using the Fourier–Galerkin method for the azimuthal direction has two advantages. Firstly, it has a high-order approximation of the partial derivatives in the angular direction, and secondly, it naturally satisfies the azimuthal periodic boundary conditions. Also, using the finite-volume method in the r and z directions allows one to handle boundary conditions with discontinuities in those directions. It is important to remark that with this method, the resulting linear system of equations are band-diagonal, leading to fast and efficient solvers. The benefits of the mixed method are illustrated with example problems. (paper)
Richter, Christiane; Kotz, Frederik; Giselbrecht, Stefan; Helmer, Dorothea; Rapp, Bastian E
2016-06-01
The fluid mechanics of microfluidics is distinctively simpler than the fluid mechanics of macroscopic systems. In macroscopic systems effects such as non-laminar flow, convection, gravity etc. need to be accounted for all of which can usually be neglected in microfluidic systems. Still, there exists only a very limited selection of channel cross-sections for which the Navier-Stokes equation for pressure-driven Poiseuille flow can be solved analytically. From these equations, velocity profiles as well as flow rates can be calculated. However, whenever a cross-section is not highly symmetric (rectangular, elliptical or circular) the Navier-Stokes equation can usually not be solved analytically. In all of these cases, numerical methods are required. However, in many instances it is not necessary to turn to complex numerical solver packages for deriving, e.g., the velocity profile of a more complex microfluidic channel cross-section. In this paper, a simple spreadsheet analysis tool (here: Microsoft Excel) will be used to implement a simple numerical scheme which allows solving the Navier-Stokes equation for arbitrary channel cross-sections.
International Nuclear Information System (INIS)
Thompson, K.G.
2000-01-01
In this work, we develop a new spatial discretization scheme that may be used to numerically solve the neutron transport equation. This new discretization extends the family of corner balance spatial discretizations to include spatial grids of arbitrary polyhedra. This scheme enforces balance on subcell volumes called corners. It produces a lower triangular matrix for sweeping, is algebraically linear, is non-negative in a source-free absorber, and produces a robust and accurate solution in thick diffusive regions. Using an asymptotic analysis, we design the scheme so that in thick diffusive regions it will attain the same solution as an accurate polyhedral diffusion discretization. We then refine the approximations in the scheme to reduce numerical diffusion in vacuums, and we attempt to capture a second order truncation error. After we develop this Upstream Corner Balance Linear (UCBL) discretization we analyze its characteristics in several limits. We complete a full diffusion limit analysis showing that we capture the desired diffusion discretization in optically thick and highly scattering media. We review the upstream and linear properties of our discretization and then demonstrate that our scheme captures strictly non-negative solutions in source-free purely absorbing media. We then demonstrate the minimization of numerical diffusion of a beam and then demonstrate that the scheme is, in general, first order accurate. We also note that for slab-like problems our method actually behaves like a second-order method over a range of cell thicknesses that are of practical interest. We also discuss why our scheme is first order accurate for truly 3D problems and suggest changes in the algorithm that should make it a second-order accurate scheme. Finally, we demonstrate 3D UCBL's performance on several very different test problems. We show good performance in diffusive and streaming problems. We analyze truncation error in a 3D problem and demonstrate robustness in a
GMES: A Python package for solving Maxwell’s equations using the FDTD method
Chun, Kyungwon; Kim, Huioon; Kim, Hyounggyu; Jung, Kil Su; Chung, Youngjoo
2013-04-01
This paper describes GMES, a free Python package for solving Maxwell’s equations using the finite-difference time-domain (FDTD) method. The design of GMES follows the object-oriented programming (OOP) approach and adopts a unique design strategy where the voxels in the computational domain are grouped and then updated according to its material type. This piecewise updating scheme ensures that GMES can adopt OOP without losing its simple structure and time-stepping speed. The users can easily add various material types, sources, and boundary conditions into their code using the Python programming language. The key design features, along with the supported material types, excitation sources, boundary conditions and parallel calculations employed in GMES are also described in detail. Catalog identifier: AEOK_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEOK_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License v3.0 No. of lines in distributed program, including test data, etc.: 17700 No. of bytes in distributed program, including test data, etc.: 89878 Distribution format: tar.gz Programming language: C++, Python. Computer: Any computer with a Unix-like system with a C++ compiler, and a Python interpreter; developed on 2.53 GHz Intel CoreTM i3. Operating system: Any Unix-like system; developed under Ubuntu 12.04 LTS 64 bit. Has the code been vectorized or parallelized?: Yes. Parallelized with MPI directives (optional). RAM: Problem dependent (a simulation with real valued electromagnetic field uses roughly 0.18 KB per Yee cell.) Classification: 10. External routines: SWIG [1], Cython [2], NumPy [3], SciPy [4], matplotlib [5], MPI for Python [6] Nature of problem: Classical electrodynamics Solution method: Finite-difference time-domain (FDTD) method Additional comments: This article describes version 0.9.5. The most recent version can be downloaded at the GMES
Tayebi, A.; Shekari, Y.; Heydari, M. H.
2017-07-01
Several physical phenomena such as transformation of pollutants, energy, particles and many others can be described by the well-known convection-diffusion equation which is a combination of the diffusion and advection equations. In this paper, this equation is generalized with the concept of variable-order fractional derivatives. The generalized equation is called variable-order time fractional advection-diffusion equation (V-OTFA-DE). An accurate and robust meshless method based on the moving least squares (MLS) approximation and the finite difference scheme is proposed for its numerical solution on two-dimensional (2-D) arbitrary domains. In the time domain, the finite difference technique with a θ-weighted scheme and in the space domain, the MLS approximation are employed to obtain appropriate semi-discrete solutions. Since the newly developed method is a meshless approach, it does not require any background mesh structure to obtain semi-discrete solutions of the problem under consideration, and the numerical solutions are constructed entirely based on a set of scattered nodes. The proposed method is validated in solving three different examples including two benchmark problems and an applied problem of pollutant distribution in the atmosphere. In all such cases, the obtained results show that the proposed method is very accurate and robust. Moreover, a remarkable property so-called positive scheme for the proposed method is observed in solving concentration transport phenomena.
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Dumitru Baleanu
2013-01-01
Full Text Available We obtain the approximate analytical solution for the fractional quadratic Riccati differential equation with the Riemann-Liouville derivative by using the Bernstein polynomials (BPs operational matrices. In this method, we use the operational matrix for fractional integration in the Riemann-Liouville sense. Then by using this matrix and operational matrix of product, we reduce the problem to a system of algebraic equations that can be solved easily. The efficiency and accuracy of the proposed method are illustrated by several examples.
Computational issues of solving the 1D steady gradually varied flow equation
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Artichowicz Wojciech
2014-09-01
Full Text Available In this paper a problem of multiple solutions of steady gradually varied flow equation in the form of the ordinary differential energy equation is discussed from the viewpoint of its numerical solution. Using the Lipschitz theorem dealing with the uniqueness of solution of an initial value problem for the ordinary differential equation it was shown that the steady gradually varied flow equation can have more than one solution. This fact implies that the nonlinear algebraic equation approximating the ordinary differential energy equation, which additionally coincides with the wellknown standard step method usually applied for computing of the flow profile, can have variable number of roots. Consequently, more than one alternative solution corresponding to the same initial condition can be provided. Using this property it is possible to compute the water flow profile passing through the critical stage.
Rahan, Nur Nadiah Mohd; Ishak, Siti Noor Shahira; Hamid, Nur Nadiah Abd; Majid, Ahmad Abd.; Azmi, Amirah
2017-04-01
In this research, the nonlinear Benjamin-Bona-Mahony (BBM) equation is solved numerically using the cubic B-spline (CuBS) and cubic trigonometric B-spline (CuTBS) collocation methods. The CuBS and CuTBS are utilized as interpolating functions in the spatial dimension while the standard finite difference method (FDM) is applied to discretize the temporal space. In order to solve the nonlinear problem, the BBM equation is linearized using Taylor's expansion. Applying the von-Neumann stability analysis, the proposed techniques are shown to be unconditionally stable under the Crank-Nicolson scheme. Several numerical examples are discussed and compared with exact solutions and results from the FDM.
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Uswah Qasim
2016-03-01
Full Text Available A homotopy method is presented for the construction of frozen Jacobian iterative methods. The frozen Jacobian iterative methods are attractive because the inversion of the Jacobian is performed in terms of LUfactorization only once, for a single instance of the iterative method. We embedded parameters in the iterative methods with the help of the homotopy method: the values of the parameters are determined in such a way that a better convergence rate is achieved. The proposed homotopy technique is general and has the ability to construct different families of iterative methods, for solving weakly nonlinear systems of equations. Further iterative methods are also proposed for solving general systems of nonlinear equations.
Dinar, N.
1978-01-01
Several aspects of multigrid methods are briefly described. The main subjects include the development of very efficient multigrid algorithms for systems of elliptic equations (Cauchy-Riemann, Stokes, Navier-Stokes), as well as the development of control and prediction tools (based on local mode Fourier analysis), used to analyze, check and improve these algorithms. Preliminary research on multigrid algorithms for time dependent parabolic equations is also described. Improvements in existing multigrid processes and algorithms for elliptic equations were studied.
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Fayyaz Ahmad
2017-01-01
Full Text Available A modification to an existing iterative method for computing zeros with unknown multiplicities of nonlinear equations or a system of nonlinear equations is presented. We introduce preconditioners to nonlinear equations or a system of nonlinear equations and their corresponding Jacobians. The inclusion of preconditioners provides numerical stability and accuracy. The different selection of preconditioner offers a family of iterative methods. We modified an existing method in a way that we do not alter its inherited quadratic convergence. Numerical simulations confirm the quadratic convergence of the preconditioned iterative method. The influence of preconditioners is clearly reflected in the numerically achieved accuracy of computed solutions.
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Mahmoud Paripour
2014-08-01
Full Text Available In this paper, the Bernstein polynomials are used to approximatethe solutions of linear integral equations with multiple time lags (IEMTL through expansion methods (collocation method, partition method, Galerkin method. The method is discussed in detail and illustrated by solving some numerical examples. Comparison between the exact and approximated results obtained from these methods is carried out
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Elhassan Eljaoui
2018-01-01
Full Text Available We introduce the Aumann fuzzy improper integral to define the convolution product of a fuzzy mapping and a crisp function in this paper. The Laplace convolution formula is proved in this case and used to solve fuzzy integro-differential equations with kernel of convolution type. Then, we report and correct an error in the article by Salahshour et al. dealing with the same topic.
shadan sadigh behzadi
2012-01-01
In this present paper, we solve a two-dimensional nonlinear Volterra-Fredholm integro-differential equation by using the following powerful, efficient but simple methods: (i) Modified Adomian decomposition method (MADM), (ii) Variational iteration method (VIM), (iii) Homotopy analysis method (HAM) and (iv) Modified homotopy perturbation method (MHPM). The uniqueness of the solution and the convergence of the proposed methods are proved in detail. Numerical examples are studied to demonstrate ...
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shadan sadigh behzadi
2012-03-01
Full Text Available In this present paper, we solve a two-dimensional nonlinear Volterra-Fredholm integro-differential equation by using the following powerful, efficient but simple methods: (i Modified Adomian decomposition method (MADM, (ii Variational iteration method (VIM, (iii Homotopy analysis method (HAM and (iv Modified homotopy perturbation method (MHPM. The uniqueness of the solution and the convergence of the proposed methods are proved in detail. Numerical examples are studied to demonstrate the accuracy of the presented methods.
Xia, Youshen; Feng, Gang; Wang, Jun
2004-09-01
This paper presents a recurrent neural network for solving strict convex quadratic programming problems and related linear piecewise equations. Compared with the existing neural networks for quadratic program, the proposed neural network has a one-layer structure with a low model complexity. Moreover, the proposed neural network is shown to have a finite-time convergence and exponential convergence. Illustrative examples further show the good performance of the proposed neural network in real-time applications.
Benhammouda, Brahim
2016-01-01
Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.
Energy Technology Data Exchange (ETDEWEB)
Sethian, J.A.; Shan, Y.
2007-12-10
We present a numerical algorithm for solving partial differential equations on irregular domains with moving interfaces. Instead of the typical approach of solving in a larger rectangular domain, our approach performs most calculations only in the desired domain. To do so efficiently, we have developed a one-sided multigrid method to solve the corresponding large sparse linear systems. Our focus is on the simulation of the electrodeposition process in semiconductor manufacturing in both two and three dimensions. Our goal is to track the position of the interface between the metal and the electrolyte as the features are filled and to determine which initial configurations and physical parameters lead to superfilling. We begin by motivating the set of equations which model the electrodeposition process. Building on existing models for superconformal electrodeposition, we develop a model which naturally arises from a conservation law form of surface additive evolution. We then introduce several numerical algorithms, including a conservative material transport level set method and our multigrid method for one-sided diffusion equations. We then analyze the accuracy of our numerical methods. Finally, we compare our result with experiment over a wide range of physical parameters.
International Nuclear Information System (INIS)
Zerr, R.J.; Azmy, Y.Y.
2010-01-01
A spatial domain decomposition with a parallel block Jacobi solution algorithm has been developed based on the integral transport matrix formulation of the discrete ordinates approximation for solving the within-group transport equation. The new methodology abandons the typical source iteration scheme and solves directly for the fully converged scalar flux. Four matrix operators are constructed based upon the integral form of the discrete ordinates equations. A single differential mesh sweep is performed to construct these operators. The method is parallelized by decomposing the problem domain into several smaller sub-domains, each treated as an independent problem. The scalar flux of each sub-domain is solved exactly given incoming angular flux boundary conditions. Sub-domain boundary conditions are updated iteratively, and convergence is achieved when the scalar flux error in all cells meets a pre-specified convergence criterion. The method has been implemented in a computer code that was then employed for strong scaling studies of the algorithm's parallel performance via a fixed-size problem in tests ranging from one domain up to one cell per sub-domain. Results indicate that the best parallel performance compared to source iterations occurs for optically thick, highly scattering problems, the variety that is most difficult for the traditional SI scheme to solve. Moreover, the minimum execution time occurs when each sub-domain contains a total of four cells. (authors)
Applications of algebraic method to exactly solve some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)]. E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)]. E-mail: aramady@yahoo.com
2007-08-15
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear evolution equations is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDE's) are obtained. Graphs of the solutions are displayed.
A Method for Solving the Voltage and Torque Equations of the Split ...
African Journals Online (AJOL)
Single phase induction machines have been the subject of many researches in recent times. The voltage and torque equations which describe the dynamic characteristics of these machines have been quoted in many papers, including the papers that present the simulation results of these model equations. The way and ...
Directory of Open Access Journals (Sweden)
N. Ahmadi
2017-02-01
Full Text Available Abstract In this paper, we present a collocation method based on Gaussian Radial Basis Functions (RBFs for approximating the solution of stochastic fractional differential equations (SFDEs. In this equation the fractional derivative is considered in the Caputo sense. Also we prove the existence and uniqueness of the presented method. Numerical examples confirm the proficiency of the method.
Dissecting zero modes and bound states on BPS vortices in Ginzburg-Landau superconductors
International Nuclear Information System (INIS)
Izquierdo, A. Alonso; Fuertes, W. Garcia; Guilarte, J. Mateos
2016-01-01
In this paper the zero modes of fluctuation of cylindrically symmetric self-dual vortices are analyzed and described in full detail. These BPS topological defects arise at the critical point between Type II and Type I superconductors, or, equivalently, when the masses of the Higgs particle and the vector boson in the Abelian Higgs model are equal. In addition, novel bound states of Higss and vector bosons trapped by the self-dual vortices at their core are found and investigated.
Dissecting zero modes and bound states on BPS vortices in Ginzburg-Landau superconductors
Energy Technology Data Exchange (ETDEWEB)
Izquierdo, A. Alonso [Departamento de Matematica Aplicada, Universidad de Salamanca,Facultad de Ciencias Agrarias y Ambientales,Av. Filiberto Villalobos 119, E-37008 Salamanca (Spain); Fuertes, W. Garcia [Departamento de Fisica, Universidad de Oviedo, Facultad de Ciencias,Calle Calvo Sotelo s/n, E-33007 Oviedo (Spain); Guilarte, J. Mateos [Departamento de Fisica Fundamental, Universidad de Salamanca, Facultad de Ciencias,Plaza de la Merced, E-37008 Salamanca (Spain)
2016-05-12
In this paper the zero modes of fluctuation of cylindrically symmetric self-dual vortices are analyzed and described in full detail. These BPS topological defects arise at the critical point between Type II and Type I superconductors, or, equivalently, when the masses of the Higgs particle and the vector boson in the Abelian Higgs model are equal. In addition, novel bound states of Higss and vector bosons trapped by the self-dual vortices at their core are found and investigated.
Ginzburg-Landau theory of superconducting surfaces under the influence of electric fields
Czech Academy of Sciences Publication Activity Database
Lipavský, Pavel; Morawetz, K.; Koláček, Jan; Yang, T.-J.
2006-01-01
Roč. 73, č. 5 (2006), 052505/1-052505/4 ISSN 1098-0121 R&D Projects: GA ČR(CZ) GA202/04/0585; GA ČR(CZ) GA202/05/0173; GA AV ČR(CZ) IAA1010312 Grant - others:National Science Consil of Taiwan (TW) NSC 94-2112-M-009-001 Institutional research plan: CEZ:AV0Z10100521 Keywords : superconductivity * thin layers Subject RIV: BE - Theoretical Physics Impact factor: 3.107, year: 2006
C1-alpha convergence of minimizers of a Ginzburg-Landau functional
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Yutian Lei
2000-02-01
Full Text Available In this article we study the minimizers of the functional $$ E_varepsilon(u,G={1over p}int_G|abla u|^p+frac{1 over 4varepsilon^p} int_G(1-|u|^2^2, $$ on the class $W_g={v in W^{1,p}(G,{mathbb R}^2;v|_{partial G}=g}$, where $g:partial G o S^1$ is a smooth map with Brouwer degree zero, and $p$ is greater than 2. In particular, we show that the minimizer converges to the $p$-harmonic map in $C_{hbox{loc}}^{1,alpha}(G,{mathbb R}^2$ as $varepsilon$ approaches zero.
Thermodynamic properties of and Nuclei using modified Ginzburg-Landau theory
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V Dehghani
2016-09-01
Full Text Available In this paper, formulation of Modified Ginsberg – Landau theory of second grade phase transitions has been expressed. Using this theory, termodynamic properties, such as heat capacity, energy, entropy and order parameters ofandnuclei has been investigated. In the heat capacity curve, calculated according to tempreture, a smooth peak is observed which is assumed to be a signature of transition from the paired phase to the normal phase of the nuclei. The same pattern is also observed in the experimental data of the heat capacity of the studied nuclei. Calculations of this model shows that, by increasing tempreture, expectation value of the order parameter tends to zero with smoother slip, comparing with Ginsberg – Landau theory. This indicates that the pairing effect exists between nucleons even at high temperatures. The experimental data obtained confirms the results of the model qualitatively.
Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel Antonio; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Marin-Hernandez, Antonio; Herrera-May, Agustin Leobardo; Diaz-Sanchez, Alejandro; Huerta-Chua, Jesus
2014-01-01
In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems are solved: three-point boundary valued problem (BVP) of third-order with a hyperbolic sine nonlinearity, two-point BVP for a second-order nonlinear differential equation with an exponential nonlinearity, and a two-point BVP for a third-order nonlinear differential equation with a radical nonlinearity. The result shows that the MTSM method is capable to generate easily computable and highly accurate approximations for nonlinear equations. 34L30.
An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation
Zhan, Ge
2013-02-19
The pure P-wave equation for modelling and migration in tilted transversely isotropic (TTI) media has attracted more and more attention in imaging seismic data with anisotropy. The desirable feature is that it is absolutely free of shear-wave artefacts and the consequent alleviation of numerical instabilities generally suffered by some systems of coupled equations. However, due to several forward-backward Fourier transforms in wavefield updating at each time step, the computational cost is significant, and thereby hampers its prevalence. We propose to use a hybrid pseudospectral (PS) and finite-difference (FD) scheme to solve the pure P-wave equation. In the hybrid solution, most of the cost-consuming wavenumber terms in the equation are replaced by inexpensive FD operators, which in turn accelerates the computation and reduces the computational cost. To demonstrate the benefit in cost saving of the new scheme, 2D and 3D reverse-time migration (RTM) examples using the hybrid solution to the pure P-wave equation are carried out, and respective runtimes are listed and compared. Numerical results show that the hybrid strategy demands less computation time and is faster than using the PS method alone. Furthermore, this new TTI RTM algorithm with the hybrid method is computationally less expensive than that with the FD solution to conventional TTI coupled equations. © 2013 Sinopec Geophysical Research Institute.
A stepwise Cas course for solving First-Order Partial Diferential Equations
Aguilera-Venegas, Gabriel; Galán-García, José Luis; Galán-García, María Ángeles; Padilla-Domínguez, Yolanda; Rodríguez-Cielos, Pedro; Rodríguez-Cielos, Ricardo
2014-01-01
Partial Differential Equations (PDE) is a very important topic in advance Mathematics for Engineering. The three main first-order PDE problems that a basic course must deal with are: \\begin{enumerate} \\item {\\bf Pfaff Differential Equations}, which consists on finding the general solution for: $$ P(x,y,z)\\:{\\rm d}x + Q(x,y,z)\\:{\\rm d}y + Q(x,y,z)\\:{\\rm d}z = 0 $$ \\item {\\bf Quasi-linear Partial Differential Equations}, which consists on finding the general solution for: ...
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Guo Zheng-Hong
2016-01-01
Full Text Available In this article, the Sumudu transform series expansion method is used to handle the local fractional Laplace equation arising in the steady fractal heat-transfer problem via local fractional calculus.
Assanova, Anar T.; Bakirova, Elmira A.; Kadirbayeva, Zhazira M.
2017-09-01
The periodic problem for a system of integro-differential equations of hyperbolic type with impulse effects is considered. This problem is reduced to an equivalent problem, consisting of a family of periodic boundary value problems for system of ordinary differential equations with parameter and impulse effects and integral relations by method of introducing additional functions. Sufficient conditions for existence of unique solution to the family of periodic boundary value problem with the impulse effects for system of the ordinary differential equations are received by parametrization method. Conditions for the unique solvability of periodic problem for system of integro - differential equations of hyperbolic type with impulse effects are established in the term of initial data.
Spinor-electron wave guided modes in coupled quantum wells structures by solving the Dirac equation
Energy Technology Data Exchange (ETDEWEB)
Linares, Jesus [Area de Optica, Departamento de Fisica Aplicada, Facultade de Fisica, Escola Universitaria de Optica e Optometria, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia (Spain)], E-mail: suso.linares.beiras@usc.es; Nistal, Maria C. [Area de Optica, Departamento de Fisica Aplicada, Facultade de Fisica, Escola Universitaria de Optica e Optometria, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia (Spain)
2009-05-04
A quantum analysis based on the Dirac equation of the propagation of spinor-electron waves in coupled quantum wells, or equivalently coupled electron waveguides, is presented. The complete optical wave equations for Spin-Up (SU) and Spin-Down (SD) spinor-electron waves in these electron guides couplers are derived from the Dirac equation. The relativistic amplitudes and dispersion equations of the spinor-electron wave-guided modes in a planar quantum coupler formed by two coupled quantum wells, or equivalently by two coupled slab electron waveguides, are exactly derived. The main outcomes related to the spinor modal structure, such as the breaking of the non-relativistic degenerate spin states, the appearance of phase shifts associated with the spin polarization and so on, are shown.
New Quasi-Newton Method for Solving Systems of Nonlinear Equations
Czech Academy of Sciences Publication Activity Database
Lukšan, Ladislav; Vlček, Jan
2017-01-01
Roč. 62, č. 2 (2017), s. 121-134 ISSN 0862-7940 R&D Projects: GA ČR GA13-06684S Institutional support: RVO:67985807 Keywords : nonlinear equations * systems of equations * trust-region methods * quasi- Newton methods * adjoint Broyden methods * numerical algorithms * numerical experiments Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.618, year: 2016 http://hdl.handle.net/10338.dmlcz/146699
Development of Galerkin Method for Solving the Generalized Burger's-Huxley Equation
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M. El-Kady
2013-01-01
Full Text Available Numerical treatments for the generalized Burger's—Huxley GBH equation are presented. The treatments are based on cardinal Chebyshev and Legendre basis functions with Galerkin method. Gauss quadrature formula and El-gendi method are used to convert the problem into a system of ordinary differential equations. The numerical results are compared with the literatures to show efficiency of the proposed methods.
Analyses of glass transition phenomena by solving differential equation with delay effect
International Nuclear Information System (INIS)
Takeuchi, A.; Inoue, A.
2007-01-01
A linear differential equation for the analyses of glass transition phenomena has been proposed by taking into account the delay effect due to the change in transportation of atoms near the glass transition temperature (T g ). Under the condition maintaining the order of the differential equation as the second, the non-linear differential equation proposed by Van Den Beukel and Sietsma is modified to obtain the analytic solution for a linear equation by introducing the following points: the delay effect which is described with a term of Mackey-Glass model, a concept of effective free volume (x fe eff ) and its concentration expression (C fe eff ) which correspond to the equilibrium, and an additional term associated with C fe eff . In analyzing the linear equation, Doyle's p-function was used for the integral of reaction rate with respect to temperature (T). It is found that the linear equation proposed in the present study can describe the changes in free volume (x) with increasing temperature in the dx/dT-T chart, the sharp increase in free volume at T g , and over shooting phenomena of free volume slightly above the T g , as experimentally in thermal analyses for metallic glasses. The linear solution obtained in the present study is of great importance for the analyses of the glass transition because the change in free volume with increasing temperature on heating is described with fundamental functions
Yanti, Y. R.; Amin, S. M.; Sulaiman, R.
2018-01-01
This study described representation of students who have musical, logical-mathematic and naturalist intelligence in solving a problem. Subjects were selected on the basis of multiple intelligence tests (TPM) consists of 108 statements, with 102 statements adopted from Chislet and Chapman and 6 statements equal to eksistensial intelligences. Data were analyzed based on problem-solving tests (TPM) and interviewing. See the validity of the data then problem-solving tests (TPM) and interviewing is given twice with an analyzed using the representation indikator and the problem solving step. The results showed that: the stage of presenting information known, stage of devising a plan, and stage of carrying out the plan those three subjects were using same form of representation. While he stage of presenting information asked and stage of looking back, subject of logical-mathematic was using different forms of representation with subjects of musical and naturalist intelligence. From this research is expected to provide input to the teacher in determining the learning strategy that will be used by considering the representation of students with the basis of multiple intelligences.
A Distributed Secure Outsourcing Scheme for Solving Linear Algebraic Equations in Ad Hoc Clouds
Shen, Wenlong; Yin, Bo; Cao, Xianghui; Cheng, Yu
2015-01-01
The emerging ad hoc clouds form a new cloud computing paradigm by leveraging untapped local computation and storage resources. An important application application over ad hoc clouds is outsourcing computationally intensive problems to nearby cloud agents to solve in a distributed manner.
International Nuclear Information System (INIS)
Fogelson, A.L.; Peskin, C.S.
1988-01-01
A new fast numerical method for solving the three-dimensional Stokes' equations in the presence of suspended particles is presented. The fluid dynamics equations are solved on a lattice. A particle is represented by a set of points each of which moves at the local fluid velocity and is not constrained to lie on the lattice. These points are coupled by forces which resist deformation of the particle. These forces contribute to the force density in the Stokes' equations. As a result, a single set of fluid dynamics equations holds at all points of the domain and there are no internal boundaries. Particles size, shape, and deformability may be prescribed. Computational work increases only linearly with the number of particles, so large numbers (500--1000) of particles may be studied efficiently. The numerical method involves implicit calculation of the particle forces by minimizing an energy function and solution of a finite-difference approximation to the Stokes' equations using the Fourier--Toeplitz method. The numerical method has been implemented to run on all CRAY computers: the implementation exploits the CRAY's vectorized arithmetic, and on machines with insufficient central memory, it performs efficient disk I/O while storing most of the data on disk. Applications of the method to sedimentation of one-, two-, and many-particle systems are described. Trajectories and settling speeds for two-particle sedimentation, and settling speed for multiparticle sedimentation from initial distributions on a cubic lattice or at random give good quantitative agreement with existing theories. copyright 1988 Academic Press, Inc
An ansatz for solving nonlinear partial differential equations in mathematical physics.
Akbar, M Ali; Ali, Norhashidah Hj Mohd
2016-01-01
In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.
Comparative analysis among several methods used to solve the point kinetic equations
International Nuclear Information System (INIS)
Nunes, Anderson L.; Goncalves, Alessandro da C.; Martinez, Aquilino S.; Silva, Fernando Carvalho da
2007-01-01
The main objective of this work consists on the methodology development for comparison of several methods for the kinetics equations points solution. The evaluated methods are: the finite differences method, the stiffness confinement method, improved stiffness confinement method and the piecewise constant approximations method. These methods were implemented and compared through a systematic analysis that consists basically of confronting which one of the methods consume smaller computational time with higher precision. It was calculated the relative which function is to combine both criteria in order to reach the goal. Through the analyses of the performance factor it is possible to choose the best method for the solution of point kinetics equations. (author)
Comparative analysis among several methods used to solve the point kinetic equations
Energy Technology Data Exchange (ETDEWEB)
Nunes, Anderson L.; Goncalves, Alessandro da C.; Martinez, Aquilino S.; Silva, Fernando Carvalho da [Universidade Federal, Rio de Janeiro, RJ (Brazil). Coordenacao dos Programas de Pos-graduacao de Engenharia. Programa de Engenharia Nuclear; E-mails: alupo@if.ufrj.br; agoncalves@con.ufrj.br; aquilino@lmp.ufrj.br; fernando@con.ufrj.br
2007-07-01
The main objective of this work consists on the methodology development for comparison of several methods for the kinetics equations points solution. The evaluated methods are: the finite differences method, the stiffness confinement method, improved stiffness confinement method and the piecewise constant approximations method. These methods were implemented and compared through a systematic analysis that consists basically of confronting which one of the methods consume smaller computational time with higher precision. It was calculated the relative which function is to combine both criteria in order to reach the goal. Through the analyses of the performance factor it is possible to choose the best method for the solution of point kinetics equations. (author)
Solving Matrix Equations on Multi-Core and Many-Core Architectures
Directory of Open Access Journals (Sweden)
Peter Benner
2013-11-01
Full Text Available We address the numerical solution of Lyapunov, algebraic and differential Riccati equations, via the matrix sign function, on platforms equipped with general-purpose multicore processors and, optionally, one or more graphics processing units (GPUs. In particular, we review the solvers for these equations, as well as the underlying methods, analyze their concurrency and scalability and provide details on their parallel implementation. Our experimental results show that this class of hardware provides sufficient computational power to tackle large-scale problems, which only a few years ago would have required a cluster of computers.
Generalized Kudryashov method for solving some (3+1-dimensional nonlinear evolution equations
Directory of Open Access Journals (Sweden)
Md. Shafiqul Islam
2015-06-01
Full Text Available In this work, we have applied the generalized Kudryashov methods to obtain the exact travelling wave solutions for the (3+1-dimensional Jimbo-Miwa (JM equation, the (3+1-dimensional Kadomtsev-Petviashvili (KP equation and the (3+1-dimensional Zakharov-Kuznetsov (ZK. The attained solutions show distinct physical configurations. The constraints that will guarantee the existence of specific solutions will be investigated. These solutions may be useful and desirable for enlightening specific nonlinear physical phenomena in genuinely nonlinear dynamical systems.
Solving the incompressible surface Navier-Stokes equation by surface finite elements
Reuther, Sebastian; Voigt, Axel
2018-01-01
We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus g (S ) . The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding R3, penalization of the normal component, a Chorin projection method, and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincaré-Hopf theorem on n-tori.
NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION
Liu, F.; Meerschaert, M.M.; McGough, R.J.; Zhuang, P.; Liu, Q.
2013-01-01
In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and technique...
Johnson, Ryan; Kercher, Andrew; Schwer, Douglas; Corrigan, Andrew; Kailasanath, Kazhikathra
2017-11-01
This presentation focuses on the development of a Discontinuous Galerkin (DG) method for application to chemically reacting flows. The in-house code, called Propel, was developed by the Laboratory of Computational Physics and Fluid Dynamics at the Naval Research Laboratory. It was designed specifically for developing advanced multi-dimensional algorithms to run efficiently on new and innovative architectures such as GPUs. For these results, Propel solves for convection and diffusion simultaneously with detailed transport and thermodynamics. Chemistry is currently solved in a time-split approach using Strang-splitting with finite element DG time integration of chemical source terms. Results presented here show canonical unsteady reacting flow cases, such as co-flow and splitter plate, and we report performance for higher order DG on CPU and GPUs.
International Nuclear Information System (INIS)
Verdu, G.; Miro, R.; Ginestar, D.; Vidal, V.
1999-01-01
To calculate the neutronic steady state of a nuclear power reactor core and its subcritical modes, it is necessary to solve a partial eigenvalue problem. In this paper, an implicit restarted Arnoldi method is presented as an advantageous alternative to classical methods as the Power Iteration method and the Subspace Iteration method. The efficiency of these methods, has been compared calculating the dominant Lambda modes of several configurations of the Three Mile Island reactor core
Energy Technology Data Exchange (ETDEWEB)
Verdu, G.; Miro, R. [Departamento de Ingenieria Quimica y Nuclear, Universidad Politecnica de Valencia, Valencia (Spain); Ginestar, D. [Departamento de Matematica Aplicada, Universidad Politecnica de Valencia, Valencia (Spain); Vidal, V. [Departamento de Sistemas Informaticos y Computacion, Universidad Politecnica de Valencia, Valencia (Spain)
1999-05-01
To calculate the neutronic steady state of a nuclear power reactor core and its subcritical modes, it is necessary to solve a partial eigenvalue problem. In this paper, an implicit restarted Arnoldi method is presented as an advantageous alternative to classical methods as the Power Iteration method and the Subspace Iteration method. The efficiency of these methods, has been compared calculating the dominant Lambda modes of several configurations of the Three Mile Island reactor core.
Piret, Cécile
2012-05-01
Much work has been done on reconstructing arbitrary surfaces using the radial basis function (RBF) method, but one can hardly find any work done on the use of RBFs to solve partial differential equations (PDEs) on arbitrary surfaces. In this paper, we investigate methods to solve PDEs on arbitrary stationary surfaces embedded in . R3 using the RBF method. We present three RBF-based methods that easily discretize surface differential operators. We take advantage of the meshfree character of RBFs, which give us a high accuracy and the flexibility to represent the most complex geometries in any dimension. Two out of the three methods, which we call the orthogonal gradients (OGr) methods are the result of our work and are hereby presented for the first time. © 2012 Elsevier Inc.
International Nuclear Information System (INIS)
Matausek, M.
1972-01-01
A new proposed method for solving the space-energy dependent spherical harmonics equations represents a methodological contribution to neutron transport theory. The proposed method was applied for solving the problem of spec-energy transport of fast and resonance neutrons in multi-zone, cylindrical y symmetric infinite reactor cell and is related to previously developed procedure for treating the thermal energy region. The advantages of this method are as follows: a unique algorithm was obtained for detailed determination of spatial and energy distribution of neutrons (from thermal to fast) in the reactor cell; these detailed distributions enable more precise calculations of criticality conditions, obtaining adequate multigroup data and better interpretation of experimental data; computing time is rather short
Solving Algebraic Riccati Equation Real Time for Integrated Vehicle Dynamics Control
Kunnappillil Madhusudhanan, A; Corno, M.; Bonsen, B.; Holweg, E.
2012-01-01
In this paper we present a comparison study of different computational methods to implement State Dependent Riccati Equation (SDRE) based control in real time for a vehicle dynamics control application. Vehicles are mechatronic systems with nonlinear dynamics. One of the promising nonlinear control
COMPUTATIONAL EXPERIENCE IN SOLVING LARGE LINEAR MATRIX EQUATIONS FOR AUTOMATIC CONTROL
Directory of Open Access Journals (Sweden)
Vasile Sima
2004-12-01
Full Text Available State-of-the-art, uni-processor linear matrix equation solvers for automatic control computations are investigated and compared for various problem sizes. Generalpurpose SLICOT solvers are the most efficient ones for small-size problems, but they cannot compete for larger problems with specialized solvers designed for certain problem classes.
On solving the Schrödinger equation for a complex deictic potential ...
Indian Academy of Sciences (India)
form solutions of the Schrödinger equation for an even power complex deictic potential and its variant in one dimension. For this purpose, extended complex phase-space approach is utilized and nature of the eigenvalue and the corresponding ...
New approach to solve fully fuzzy system of linear equations using ...
Indian Academy of Sciences (India)
Otadi & Mosleh (2012) have applied a linear programming approach to find the non-negative solution of a fully fuzzy matrix equation whose elements of the coefficient matrix are considered as arbitrary triangular fuzzy numbers. There are no restrictions about the elements of the coefficient matrix of the corresponding system ...
WKB: an interactive code for solving differential equations using phase integral methods
International Nuclear Information System (INIS)
White, R.B.
1978-01-01
A small code for the analysis of ordinary differential equations interactively through the use of Phase Integral Methods (WKB) has been written for use on the DEC 10. This note is a descriptive manual for those interested in using the code
On solving the Schrödinger equation for a complex deictic potential ...
Indian Academy of Sciences (India)
Making use of an ansatz for the eigenfunction, we investigate closed-form solutions of the Schrödinger equation for an even power complex deictic potential and its variant in one dimension. For this purpose, extended complex phase-space approach is utilized and nature of the eigenvalue and the corresponding ...
Approximate method for solving the velocity dependent transport equation in a slab lattice
International Nuclear Information System (INIS)
Ferrari, A.
1966-01-01
A method is described that is intended to provide an approximate solution of the transport equation in a medium simulating a water-moderated plate filled reactor core. This medium is constituted by a periodic array of water channels and absorbing plates. The velocity dependent transport equation in slab geometry is included. The computation is performed in a water channel: the absorbing plates are accounted for by the boundary conditions. The scattering of neutrons in water is assumed isotropic, which allows the use of a double Pn approximation to deal with the angular dependence. This method is able to represent the discontinuity of the angular distribution at the channel boundary. The set of equations thus obtained is dependent only on x and v and the coefficients are independent on x. This solution suggests to try solutions involving Legendre polynomials. This scheme leads to a set of equations v dependent only. To obtain an explicit solution, a thermalization model must now be chosen. Using the secondary model of Cadilhac a solution of this set is easy to get. The numerical computations were performed with a particular secondary model, the well-known model of Wigner and Wilkins. (author) [fr
A Method for Solving the Voltage and Torque Equations of the Split ...
African Journals Online (AJOL)
Akorede
acceleration characteristics of the motor voltages, currents and electromagnetic torque have been plotted and discussed. The simulation results presented include the instantaneous torque-speed characteristics of the Split phase Induction machine. A block diagram of the method for the solution of the machine equations ...
Parallel Algorithm for Solving TOV Equations for Sequence of Cold and Dense Nuclear Matter Models
Ayriyan, Alexander; Buša, Ján; Grigorian, Hovik; Poghosyan, Gevorg
2018-04-01
We have introduced parallel algorithm simulation of neutron star configurations for set of equation of state models. The performance of the parallel algorithm has been investigated for testing set of EoS models on two computational systems. It scales when using with MPI on modern CPUs and this investigation allowed us also to compare two different types of computational nodes.
Ngu, Bing Hiong; Yeung, Alexander Seeshing
2013-01-01
Text editing directs students' attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three…
ICM: an Integrated Compartment Method for numerically solving partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Yeh, G.T.
1981-05-01
An integrated compartment method (ICM) is proposed to construct a set of algebraic equations from a system of partial differential equations. The ICM combines the utility of integral formulation of finite element approach, the simplicity of interpolation of finite difference approximation, and the flexibility of compartment analyses. The integral formulation eases the treatment of boundary conditions, in particular, the Neumann-type boundary conditions. The simplicity of interpolation provides great economy in computation. The flexibility of discretization with irregular compartments of various shapes and sizes offers advantages in resolving complex boundaries enclosing compound regions of interest. The basic procedures of ICM are first to discretize the region of interest into compartments, then to apply three integral theorems of vectors to transform the volume integral to the surface integral, and finally to use interpolation to relate the interfacial values in terms of compartment values to close the system. The Navier-Stokes equations are used as an example of how to derive the corresponding ICM alogrithm for a given set of partial differential equations. Because of the structure of the algorithm, the basic computer program remains the same for cases in one-, two-, or three-dimensional problems.
The foam drainage equation with time- and space-fractional derivatives solved by the Adomian method
Directory of Open Access Journals (Sweden)
Zoubir Dahmani
2008-10-01
Full Text Available In this paper, by introducing the fractional derivative in the sense of Caputo, we apply the Adomian decomposition method for the foam drainage equation with time- and space-fractional derivative. As a result, numerical solutions are obtained in a form of rapidly convergent series with easily computable components.
Solving the Fokker-Planck equation on a massively parallel computer
International Nuclear Information System (INIS)
Mirin, A.A.
1990-01-01
The Fokker-Planck package FPPAC had been converted to the Connection Machine 2 (CM2). For fine mesh cases the CM2 outperforms the Cray-2 when it comes to time-integrating the difference equations. For long Legendre expansions the CM2 is also faster at computing the Fokker-Planck coefficients. 3 refs
Mulac, Richard A.; Celestina, Mark L.; Adamczyk, John J.; Misegades, Kent P.; Dawson, Jef M.
1987-01-01
A procedure is outlined which utilizes parallel processing to solve the inviscid form of the average-passage equation system for multistage turbomachinery along with a description of its implementation in a FORTRAN computer code, MSTAGE. A scheme to reduce the central memory requirements of the program is also detailed. Both the multitasking and I/O routines referred to in this paper are specific to the Cray X-MP line of computers and its associated SSD (Solid-state Storage Device). Results are presented for a simulation of a two-stage rocket engine fuel pump turbine.
Mulac, Richard A.; Celestina, Mark L.; Adamczyk, John J.; Misegades, Kent P.; Dawson, Jef M.
1987-01-01
A procedure is outlined which utilizes parallel processing to solve the inviscid form of the average-passage equation system for multistage turbomachinery along with a description of its implementation in a FORTRAN computer code, MSTAGE. A scheme to reduce the central memory requirements of the program is also detailed. Both the multitasking and I/O routines referred to are specific to the Cray X-MP line of computers and its associated SSD (Solid-State Disk). Results are presented for a simulation of a two-stage rocket engine fuel pump turbine.
Directory of Open Access Journals (Sweden)
Imtiaz Wasim
2018-01-01
Full Text Available In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. This technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. The scheme is verified unconditionally stable using the Von Neumann (Fourier method. Several test problems are considered to check the accuracy of the proposed scheme. The numerical results are in good agreement with known exact solutions and the existing schemes in literature.
Martin, Shirley A; Bassok, Miriam
2005-04-01
Mathematical solutions to textbook word problems are correlated with semantic relations between the objects described in the problem texts. In particular, division problems usually involve functionally related objects (e.g., tulips-vases) and rarely involve categorically related objects (e.g., tulips-daisies). We examined whether middle school, high school, and college students use object relations when they solve division word problems (WP) or perform the less familiar task of representing verbal statements with algebraic equations (EQ). Both tasks involved multiplicative comparison statements with either categorically or functionally related objects (e.g., "four times as many cupcakes [commuters] as brownies [automobiles]"). Object relations affected the frequency of correct solutions in the WP task but not in the EQ task. In the latter task, object relations did affect the structure of nonalgebraic equation errors. We argue that students use object relations as "semantic cues" when they engage in the sense-making activity of mathematical modeling.
Ahmad, Azhar; Azmi, Amirah; Majid, Ahmad Abd.; Hamid, Nur Nadiah Abd
2017-08-01
In this paper, Nonlinear Schrödinger (NLS) equation with Neumann boundary conditions is solved using finite difference method (FDM) and cubic B-spline interpolation method (CuBSIM). First, the approach is based on the FDM applied on the time and space discretization with the help of theta-weighted method. However, our main interest is the second approach, whereby FDM is applied on the time discretization and cubic B-spline is utilized as an interpolation function in the space dimension with the same help of theta-weighted method. The CuBSIM is shown to be stable by using von Neumann stability analysis. The proposed method is tested on a test problem with single soliton motion of the NLS equation. The accuracy of the numerical results is measured by the Euclidean-norm and infinity-norm. CuBSIM is found to produce more accurate results than the FDM.
Barnett, Alan R.; Ibrahim, Omar M.; Abdallah, Ayman A.; Sullivan, Timothy L.
1993-05-01
By utilizing MSC/NASTRAN DMAP (Direct Matrix Abstraction Program) in an existing NASA Lewis Research Center coupled loads methodology, solving modal equations of motion with initial conditions is possible using either coupled (Newmark-Beta) or uncoupled (exact mode superposition) integration available within module TRD1. Both the coupled and newly developed exact mode superposition methods have been used to perform transient analyses of various space systems. However, experience has shown that in most cases, significant time savings are realized when the equations of motion are integrated using the uncoupled solver instead of the coupled solver. Through the results of a real-world engineering analysis, advantages of using the exact mode superposition methodology are illustrated.
Barnett, Alan R.; Ibrahim, Omar M.; Abdallah, Ayman A.; Sullivan, Timothy L.
1993-01-01
By utilizing MSC/NASTRAN DMAP (Direct Matrix Abstraction Program) in an existing NASA Lewis Research Center coupled loads methodology, solving modal equations of motion with initial conditions is possible using either coupled (Newmark-Beta) or uncoupled (exact mode superposition) integration available within module TRD1. Both the coupled and newly developed exact mode superposition methods have been used to perform transient analyses of various space systems. However, experience has shown that in most cases, significant time savings are realized when the equations of motion are integrated using the uncoupled solver instead of the coupled solver. Through the results of a real-world engineering analysis, advantages of using the exact mode superposition methodology are illustrated.
International Nuclear Information System (INIS)
Milenkovic, B.; Stevanovic, N.; Krstic, D.
2008-01-01
The general equation of the track wall was solved numerically by using finite difference method and computer software MATHEMATICA. This method was applied for alpha particle tracks in LR115 detector, assuming both directions of etching, from the top and from the bottom of the sensitive layer. The equation of the track wall etched in reverse direction was derived, and has the same form as one for direct etching, with some difference in argument of V function. It has been shown that tracks diameter are larger in reverse etching when the energy is large and removed layer is relatively small. Opposite to this, tracks diameter are smaller in reverse etching when energy of alpha particle is less then 2 MeV. If removed layer is large both kinds of etching would produce tracks similar in size, but the track profile is different. (author)
Direct application of Padé approximant for solving nonlinear differential equations.
Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Garcia-Gervacio, Jose Luis; Huerta-Chua, Jesus; Morales-Mendoza, Luis Javier; Gonzalez-Lee, Mario
2014-01-01
This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant. 34L30.
International Nuclear Information System (INIS)
Fernandes, A.
1991-01-01
A method to solve three dimensional neutron transport equation and it is based on the original work suggested by J.K. Fletcher (42, 43). The angular dependence of the flux is approximated by associated Legendre functions and the finite element method is applied to the space components is presented. When the angular flux, the scattering cross section and the neutrons source are expanded in associated Legendre functions, the first order neutron transport equation is reduced to a coupled set of second order diffusion like equations. These equations are solved in an iterative way by the finite element method to the moments. (author)
Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation
Directory of Open Access Journals (Sweden)
Sergey V. Ershkov
2017-05-01
Full Text Available The main objective for this research was the analytical exploration of the dynamics of planar satellite rotation during the motion of an elliptical orbit around a planet. First, we revisit the results of J. Wisdom et al. (1984, in which, by the elegant change of variables (considering the true anomaly f as the independent variable, the governing equation of satellite rotation takes the form of an Abel ordinary differential equation (ODE of the second kind, a sort of generalization of the Riccati ODE. We note that due to the special character of solutions of a Riccati-type ODE, there exists the possibility of sudden jumping in the magnitude of the solution at some moment of time. In the physical sense, this jumping of the Riccati-type solutions of the governing ODE could be associated with the effect of sudden acceleration/deceleration in the satellite rotation around the chosen principle axis at a definite moment of parametric time. This means that there exists not only a chaotic satellite rotation regime (as per the results of J. Wisdom et al. (1984, but a kind of gradient catastrophe (Arnold, 1992 could occur during the satellite rotation process. We especially note that if a gradient catastrophe could occur, this does not mean that it must occur: such a possibility depends on the initial conditions. In addition, we obtained asymptotical solutions that manifest a quasi-periodic character even with the strong simplifying assumptions e→0, p=1, which reduce the governing equation of J. Wisdom et al. (1984 to a kind of Beletskii’s equation.
Solving finite-difference equations for diffractive optics problems using graphics processing units
Golovashkin, Dimitry Lvovich; Vorotnokova, Daria G.; Kochurov, Alexander V.; Malysheva, Svetlana A.
2013-09-01
This article presents a vector algorithm of the solution of the Helmholtz equation by beam propagation method (BPM), based on representation of calculations in the single instruction, multiple data model. Use of this algorithm on the graphic NVIDIA GeForce GTX 660Ti processor on the compute unified device architecture technology results in 30-times acceleration of calculations in comparison with calculations on the Intel Core2 Duo CPU E8500.
A nodal method for solving the time-depending diffusion equation in the IQS approximation
International Nuclear Information System (INIS)
Vidovsky, I.; Kereszturi, A.
1991-11-01
The fast and slow variation of the neutron flux shape needed for the dynamical description of nuclear reactor cores can be described advantageously in the Improved Quasistatic (IQS) model where the flux is factorized by a fast changing space-independent amplitude and a slow changing shape function. The basic equations of a time-dependent nodal approximation using the IQS method is presented.The calculational procedure of the response matrices is also described. (R.P.) 2 refs
A multiresolution method for solving the Poisson equation using high order regularization
DEFF Research Database (Denmark)
Hejlesen, Mads Mølholm; Walther, Jens Honore
2016-01-01
and regularized Green's functions corresponding to the difference in the spatial resolution between the patches. The full solution is obtained utilizing the linearity of the Poisson equation enabling super-position of solutions. We show that the multiresolution Poisson solver produces convergence rates......We present a novel high order multiresolution Poisson solver based on regularized Green's function solutions to obtain exact free-space boundary conditions while using fast Fourier transforms for computational efficiency. Multiresolution is a achieved through local refinement patches...
Discrete maximum principle for Poisson equation with mixed boundary conditions solved by hp-FEM
Czech Academy of Sciences Publication Activity Database
Vejchodský, Tomáš; Šolín, P.
2009-01-01
Roč. 1, č. 2 (2009), s. 201-214 ISSN 2070-0733 R&D Projects: GA AV ČR IAA100760702; GA ČR(CZ) GA102/07/0496; GA ČR GA102/05/0629 Institutional research plan: CEZ:AV0Z10190503 Keywords : discrete maximum principle * hp-FEM * Poisson equation * mixed boundary conditions Subject RIV: BA - General Mathematics
Modified exponential based differential quadrature scheme to solve convection diffusion equation
Arora, Geeta; Kataria, Pooja
2017-07-01
This paper proffers differential quadrature scheme to obtain approximate solution of one dimensional advection diffusion equation with Dirichlet's boundary conditions. The scheme uses modified exponential cubic spline basis functions to obtain the numerical results. The method uses less computational effort and produces more accurate results. In the numerical problems, L∞ and L2 errors show the relative performance of the method for different time levels. The results shown by the method are in good approximation with the exact solution.
Why Professor Richard Feynman was upset solving the Laplace equation for spherical waves?
Khelashvili, Anzor; Nadareishvili, Teimuraz
2013-01-01
We take attention to the singular behavior of the Laplace operator in spherical coordinates, which was established in our earlier work. This singularity has many non-trivial consequences. In this article we consider only the simplest ones, which are connected to the solution of Laplace equation in Feynman classical books and Lectures. Feynman was upset looking in his derived solutions, which have a fictitious singular behavior at the origin. We show how these inconsistencies can be avoided.
Homotopy decomposition method for solving one-dimensional time-fractional diffusion equation
Abuasad, Salah; Hashim, Ishak
2018-04-01
In this paper, we present the homotopy decomposition method with a modified definition of beta fractional derivative for the first time to find exact solution of one-dimensional time-fractional diffusion equation. In this method, the solution takes the form of a convergent series with easily computable terms. The exact solution obtained by the proposed method is compared with the exact solution obtained by using fractional variational homotopy perturbation iteration method via a modified Riemann-Liouville derivative.
Spectral bisection algorithm for solving Schrodinger equation using upper and lower solutions
Directory of Open Access Journals (Sweden)
Qutaibeh Deeb Katatbeh
2007-10-01
Full Text Available This paper establishes a new criteria for obtaining a sequence of upper and lower bounds for the ground state eigenvalue of Schr"odinger equation $ -Deltapsi(r+V(rpsi(r=Epsi(r$ in $N$ spatial dimensions. Based on this proposed criteria, we prove a new comparison theorem in quantum mechanics for the ground state eigenfunctions of Schrodinger equation. We determine also lower and upper solutions for the exact wave function of the ground state eigenfunctions using the computed upper and lower bounds for the eigenvalues obtained by variational methods. In other words, by using this criteria, we prove that the substitution of the lower(upper bound of the eigenvalue in Schrodinger equation leads to an upper(lower solution. Finally, two proposed iteration approaches lead to an exact convergent sequence of solutions. The first one uses Raielgh-Ritz theorem. Meanwhile, the second approach uses a new numerical spectral bisection technique. We apply our results for a wide class of potentials in quantum mechanics such as sum of power-law potentials in quantum mechanics.
Energy Technology Data Exchange (ETDEWEB)
Le Hardy, D. [Université de Nantes, LTN UMR CNRS 6607 (France); Favennec, Y., E-mail: yann.favennec@univ-nantes.fr [Université de Nantes, LTN UMR CNRS 6607 (France); Rousseau, B. [Université de Nantes, LTN UMR CNRS 6607 (France); Hecht, F. [Sorbonne Universités, UPMC Université Paris 06, UMR 7598, inria de Paris, Laboratoire Jacques-Louis Lions, F-75005, Paris (France)
2017-04-01
The contribution of this paper relies in the development of numerical algorithms for the mathematical treatment of specular reflection on borders when dealing with the numerical solution of radiative transfer problems. The radiative transfer equation being integro-differential, the discrete ordinates method allows to write down a set of semi-discrete equations in which weights are to be calculated. The calculation of these weights is well known to be based on either a quadrature or on angular discretization, making the use of such method straightforward for the state equation. Also, the diffuse contribution of reflection on borders is usually well taken into account. However, the calculation of accurate partition ratio coefficients is much more tricky for the specular condition applied on arbitrary geometrical borders. This paper presents algorithms that calculate analytically partition ratio coefficients needed in numerical treatments. The developed algorithms, combined with a decentered finite element scheme, are validated with the help of comparisons with analytical solutions before being applied on complex geometries.
A new approach for solving the three-dimensional steady Euler equations. I - General theory
Chang, S.-C.; Adamczyk, J. J.
1986-01-01
The present iterative procedure combines the Clebsch potentials and the Munk-Prim (1947) substitution principle with an extension of a semidirect Cauchy-Riemann solver to three dimensions, in order to solve steady, inviscid three-dimensional rotational flow problems in either subsonic or incompressible flow regimes. This solution procedure can be used, upon discretization, to obtain inviscid subsonic flow solutions in a 180-deg turning channel. In addition to accurately predicting the behavior of weak secondary flows, the algorithm can generate solutions for strong secondary flows and will yield acceptable flow solutions after only 10-20 outer loop iterations.
Kudryavtsev, O.; Rodochenko, V.
2018-03-01
We propose a new general numerical method aimed to solve integro-differential equations with variable coefficients. The problem under consideration arises in finance where in the context of pricing barrier options in a wide class of stochastic volatility models with jumps. To handle the effect of the correlation between the price and the variance, we use a suitable substitution for processes. Then we construct a Markov-chain approximation for the variation process on small time intervals and apply a maturity randomization technique. The result is a system of boundary problems for integro-differential equations with constant coefficients on the line in each vertex of the chain. We solve the arising problems using a numerical Wiener-Hopf factorization method. The approximate formulae for the factors are efficiently implemented by means of the Fast Fourier Transform. Finally, we use a recurrent procedure that moves backwards in time on the variance tree. We demonstrate the convergence of the method using Monte-Carlo simulations and compare our results with the results obtained by the Wiener-Hopf method with closed-form expressions of the factors.
A Divide-and-Conquer Approach for Solving Fuzzy Max-Archimedean t-Norm Relational Equations
Directory of Open Access Journals (Sweden)
Jun-Lin Lin
2014-01-01
Full Text Available A system of fuzzy relational equations with the max-Archimedean t-norm composition was considered. The relevant literature indicated that this problem can be reduced to the problem of finding all the irredundant coverings of a binary matrix. A divide-and-conquer approach is proposed to solve this problem and, subsequently, to solve the original problem. This approach was used to analyze the binary matrix and then decompose the matrix into several submatrices such that the irredundant coverings of the original matrix could be constructed using the irredundant coverings of each of these submatrices. This step was performed recursively for each of these submatrices to obtain the irredundant coverings. Finally, once all the irredundant coverings of the original matrix were found, they were easily converted into the minimal solutions of the fuzzy relational equations. Experiments on binary matrices, with the number of irredundant coverings ranging from 24 to 9680, were also performed. The results indicated that, for test matrices that could initially be partitioned into more than one submatrix, this approach reduced the execution time by more than three orders of magnitude. For the other test matrices, this approach was still useful because certain submatrices could be partitioned into more than one submatrix.
Energy Technology Data Exchange (ETDEWEB)
Bouaricha, A. [Argonne National Lab., IL (United States). Mathematics and Computer Science Div.; Schnabel, R.B. [Colorado Univ., Boulder, CO (United States). Dept. of Computer Science
1996-12-31
This paper describes a modular software package for solving systems of nonlinear equations and nonlinear least squares problems, using a new class of methods called tensor methods. It is intended for small to medium-sized problems, say with up to 100 equations and unknowns, in cases where it is reasonable to calculate the Jacobian matrix or approximate it by finite differences at each iteration. The software allows the user to select between a tensor method and a standard method based upon a linear model. The tensor method models F({ital x}) by a quadratic model, where the second-order term is chosen so that the model is hardly more expensive to form, store, or solve than the standard linear model. Moreover, the software provides two different global strategies, a line search and a two- dimensional trust region approach. Test results indicate that, in general, tensor methods are significantly more efficient and robust than standard methods on small and medium-sized problems in iterations and function evaluations.
International Nuclear Information System (INIS)
Azmy, Y.Y.; Kirk, B.L.
1990-01-01
Modern parallel computer architectures offer an enormous potential for reducing CPU and wall-clock execution times of large-scale computations commonly performed in various applications in science and engineering. Recently, several authors have reported their efforts in developing and implementing parallel algorithms for solving the neutron diffusion equation on a variety of shared- and distributed-memory parallel computers. Testing of these algorithms for a variety of two- and three-dimensional meshes showed significant speedup of the computation. Even for very large problems (i.e., three-dimensional fine meshes) executed concurrently on a few nodes in serial (nonvector) mode, however, the measured computational efficiency is very low (40 to 86%). In this paper, the authors present a highly efficient (∼85 to 99.9%) algorithm for solving the two-dimensional nodal diffusion equations on the Sequent Balance 8000 parallel computer. Also presented is a model for the performance, represented by the efficiency, as a function of problem size and the number of participating processors. The model is validated through several tests and then extrapolated to larger problems and more processors to predict the performance of the algorithm in more computationally demanding situations